2425 Physics Level N AP Course Questions (2).pdf

Physics N Chapter 2 Motion Along a Straight Line Section 2.1 Displacement, Time, and Average Velocity (1)1. A body moves along a straight line. Its position from the origin at time t is given by the equation: x = 2t2 – 8t + 5, where x is in...

Physics N Chapter 2 Motion Along a Straight Line Section 2.1 Displacement, Time, and Average Velocity (1)1. A body moves along a straight line. Its position from the origin at time t is given by the equation: x = 2t2 – 8t + 5, where x is in meters and t is in seconds. Find the average velocity of the body in the interval from t = 0 to t = 2 s. (2)2. The position-time graph of a particle is shown below. What is the average velocity of the particle in the time interval [2.0 s, 6.0 s]? Section 2.2 Instantaneous Velocity (3)3. A body moves along a straight line. Its position from the origin at time t is given by the equation: x = 2t2 – 8t + 5, where x is in meters and t is in seconds. Find the velocity of the body at t = 2 s. (4)4. The position-time graph of a particle is shown below. What is the velocity of the particle at t = 5.0 s? Level N | 1  Physics N Section 2.3 Average and Instantaneous Acceleration (5)5. A car accelerates from rest to 108 km/h in 12.0 s. What is the average acceleration of the car over these 12 seconds? (6) 6. A body moves along a straight line. Its position from the origin at time t is given by the equation: x = 2t2 – 8t + 5, where x is in meters and t is in seconds. What is the acceleration of the body at t = 2 s? (7)7. The velocity-time graph of a particle is shown below. Find the average acceleration of the particle in the interval [2.0 s, 6.0 s]. Section 2.4 Motion with Constant Acceleration (8) 8. The speed of a bus traveling due South is uniformly reduced from 54.0 km/h to 36.0 km/h in a distance of 62.5 m. a. What are the magnitude and direction of the acceleration? b. If the bus keeps decelerating at the same rate, i. how long does it take to come to rest from 54.0 km/h? ii. what distance does the bus cover before coming to a stop? (9)9. On a long straight road, a car accelerates uniformly from rest, reaching a speed of 25.0 m/s in 20.0 s. The car maintains that speed for 90.0 s while moving behind a truck. The car then accelerates uniformly to 35.0 m/s in an additional 20.0 s. After maintaining that speed for 4.00 minutes, the car uniformly decelerates at 4.00 m/s2 and comes to a stop. a. Calculate the total distance traveled by the car. b. What is the average speed of the car in the first 130 s? c. What is the average acceleration of the car in the first 130 s? Level N | 2  Physics N Section 2.5 Freely Falling Bodies (10)10. The top of a cliff is 160 m above the beach. A stone dropped from the top of the cliff falls freely from rest with a uniform acceleration of 9.80 m/s2 directed vertically downward. a. Calculate the speed with which the stone hits the beach. b. Find the time the stone needs before it hits the beach. Section 2.6 Velocity and Position by Integration (11) 11. A particle moving along a straight line starts at time t = 0 with a velocity 4.0 m/s. At any instant t (s), the acceleration of the particle is expressed by: a(t) = (6.0 m/s3)t − (8.0 m/s2). a. What is the expression of the particle’s velocity? b. Give the expression of the particle’s position. Level N | 3  Physics N Chapter 3 Motion in Two and Three Dimensions Section 3.1 Position and Velocity Vectors (12)1. The position coordinates of a moving particle are given by x = 4t − 3, y = 3t2 + 5, and z = 7, where x, y, and z are expressed in m and t in s. a. What is the position vector of the particle? b. What is the displacement between t = 1 s and t = 2 s? c. Find the magnitude of the average velocity between t = 1 s and t = 2 s. 2. The position coordinates of a moving particle are given by x = 2t3 − 5t + 3, y = (13) 3t + 4t − 5, and z = −t3 + 2t2 − 6, where x, y, and z are expressed in m and t in s. 2 a. What is the velocity of the particle at t = 1 s? b. What is the speed of the particle at t = 1 s? Section 3.2 The Acceleration Vector 3. A particle is moving with the coordinates x = 2t3 − 5t + 3, y = 3t2 + 4t − 5, and z (14) = −t3 + 2t2 − 6, where x, y, and z are expressed in m and t in s. What is its acceleration vector at t = 2 s? (15) 4. A particle is moving uniformly along a curved path. Along which direction is its acceleration vector directed? Section 3.3 Projectile Motion (16) 5. A shell is shot with a muzzle velocity of 150. m/s at an angle of 53 above the horizontal. Take g = 10.0 m/s2. a. What is the velocity vector of the shell after 3.00 s? b. Find the position vector of the shell after 3.00 s. Use the given below to answer the following three questions. A ball rolls off the edge of a tabletop 1.40 m above the floor with a horizontal velocity of 1.50 m/s, as shown in the below figure. Take g = 10.0 m/s2. Level N | 4  Physics N (17) 6. How much time does it take the ball to hit the floor? A. 0.530 s B. 0.290 s C. 0.380 s D. 0.550 s E. 0.430 s (18) 7. What is the horizontal distance covered by the ball? A. 0.600 m B. 0.700 m C. 0.800 m D. 1.20 m E. 1.40 m (19) 8. What is the velocity of the ball as it hits the ground? A. 3.70 m/s B. 6.40 m/s C. 6.70 m/s D. 5.40 m/s E. 4.90 m/s (20) 9. A gun fires a shell with a velocity of 770. m/s at an angle of 18.0 above the horizontal. By modeling the shell as a particle, find its range. Section 3.4 Motion in a Circle (21)10. A car initially traveling eastward from A turns north by traveling along a circular path ABC at a uniform speed, as shown in the diagram. The length of the arc ABC is 235 m, and the car completes the turn in 36.0 s. a. What is the magnitude of the car’s acceleration? b. What is the acceleration vector when  = 35? Level N | 5  Physics N Section 3.5 Relative Velocity (22) 11. John cycles to the North at 20.0 km/h with respect to the ground and Kate cycles to the North at 25.0 km/h. What is the velocity of Kate with respect to John? (23)12. A boat is traveling upstream at 14.0 km/h with respect to a river that is flowing at 6.00 km/h (with respect to the ground). A man runs directly across the boat, from one side to the other, at 6.00 km/h (with respect to the boat). What is the speed of the man with respect to the ground? A. 10.0 km/h B. 14.0 km/h C. 18.5 km/h D. 21.0 km/h E. 26.0 km/h Level N | 6  Physics N Chapter 4 Newton’s Laws of Motion Section 4.1 Force and Interactions (24)1. An object is subjected to the action of two forces, F1 of magnitude 5.00 N and F2 of magnitude 10.0 N, as shown below. What is the resultant of the two forces? Section 4.2 Newton’s First Law (25)2. Car A is moving at 20.0 m/s, car B at 35.0 m/s, while car C is at rest. On which of the three cars is the net force greater? Section 4.3 Newton’s Second Law (26)3. A particle of mass m1 accelerates at a1 under the action of a single constant force F. When the same force acts on a second particle of mass m2, the second particle accelerates at a2. Given that m1/m2 = 3, what is the value of the ratio a2/a1? (27)4. Two forces, one with a magnitude of 3 N and the other with a magnitude of 5 N, are acting on a box of a certain mass. For which orientations of the forces shown in the diagrams below is the magnitude of the acceleration of the box the least? Justify your answer. Level N | 7  Physics N (28) 5. [G] a. State Newton’s second law of motion. b. Under the action of a constant force of 10 N, a body moves linearly such that the dependence of its coordinate x on time t is described by the equation x = t 2 − 2t + 3 , where x is in meters and t is in seconds. Determine the mass of the body. Section 4.4 Mass and Weight (29)6. A 145 g baseball is dropped from the top of a building. Neglect the effects of air resistance and take g = 10.0 m/s2. a. What is the weight of the baseball? b. What is the acceleration of the falling ball? c. How does the net force acting on the ball vary? Level N | 8  Physics N Chapter 5 Applying Newton’s Laws Section 5.1 Using Newton’s First Law: Particles in Equilibrium (30)1. A trunk of weight 600.0 N is placed on the loading ramp of a mover’s truck. The ramp is smooth and has a slope of 35.0. What is the magnitude of the force F needed to move the trunk with a constant velocity up the ramp? (31)2. [G] A load hangs in equilibrium from two cables that form an angle of 120º, as shown below. The force of gravity acting on the load is 519 N. Determine the tension forces acting on AC and CB. (32) 3. [G] A 1,500 kg car is held in place by a light cable on a frictionless ramp. The cable makes an angle of 25° with the surface of the ramp, and the ramp itself rises at 10° above the horizontal. Find the tension in the cable and the normal reaction of the support acting on the car. Section 5.2 Using Newton’s Second Law: Dynamics of Particles (33)4. A block of mass 5.00 kg is suspended from one end of a spring attached to the ceiling of a lift. The lift accelerates upwards at a rate of 2.00 m/s2. Take g = 10.0 m/s2. a. What is the magnitude of the tension in the spring? b. If the spring is part of a spring balance calibrated in newtons, what will it read? c. The value found in b is called the apparent weight. When this quantity is equal to the real weight, what would the acceleration of the lift be? Level N | 9  Physics N d. If the apparent weight is zero, what is the direction and magnitude of the acceleration of the lift? 5. A massless inextensible string carrying two masses m1 and m2—one at each end— (34) passes through the groove of a massless pulley that rotates without friction around a horizontal axis through its center. Assume that m1 > m2 and take g = 10.0 m/s2. a. What forces act on each block? Draw the free-body diagram of each block. b. What can be said about the tensions in either part of the string? Explain. c. What can be said about the accelerations of the blocks? Explain. d. What is the acceleration of m1? (35)6. [G] From a thread thrown over a fixed pulley, two loads of masses 0.75 kg and 0.85 kg are suspended. Within 2.0 s after the start of movement, each load traveled a distance of 1.2 m. Based on experimental data, find the acceleration of gravity. Section 5.3 Friction Forces (36) 7. A person is pulling a 20.0 kg crate by means of a rope hung over his shoulder such that the rope forms an angle of 30 with the horizontal. The man pulls the rope with a force of 100 N, thereby moving the crate with a constant speed along the floor. Take g = 10.0 m/s2. a. What forces act on the crate? b. Draw the free-body diagram of the crate. c. What is the magnitude of the friction exerted on the crate? d. What is the magnitude of the normal push exerted by the surface on the crate? (37) 8. A rope is attached to a box of mass 35 kg at rest on a flat surface. The maximum force, parallel to the surface, that the rope can exert on the box before it begins to move is 200 N. What is the coefficient of static friction between the surface and the box? Take g = 10.0 m/s2. (38)9. A 75.0 kg skydiver falls through air. When he spreads his hands in the spread- eagle position, the numerical value of the constant D is 0.25 kg/m. What is the terminal velocity of the skydiver? Take g = 10.0 m/s2. Level N | 10  Physics N (39)10. [G] A 60 kg skier descends from a hill. At the bottom of the hill, the speed of the skier is 10 m/s. At that moment, the skier continues to move horizontally for 40 s before stopping. Determine the friction force and the coefficient of friction along the horizontal stretch. Take g = 10 m/s2. Section 5.4 Dynamics of Circular Motion (40)11. A car moves round a bend, which is banked at a constant angle of 10° to the horizontal. When the car is traveling at a constant speed of 18 m/s, there is no sideways frictional force on the car. The car is modeled as a particle moving in a horizontal circle of radius R meters. a. What are the forces acting on the car? b. What is the value of R? (41)12. [G] A small car of mass 0.800 kg travels at a constant speed on the inside of a track that is a vertical circle with radius 5.00 m. If the normal force exerted by the track on the car when it is at the top of the track (point B) is 6.00 N, what is the normal force on the car when it is at the bottom of the track (point A)? Take g = 10 m/s2. (42) 13. [T] A paper cup of mass m is dropped from a high location with an initial downward velocity u. The magnitude of the drag force acting on the cup is modeled by D = –kv, where k is a constant and v is the speed of the cup. a. The dot in the diagram below represents the paper cup. Complete the diagram to show the forces acting on the paper cup immediately after it has been released. b. Derive, but do not solve, the differential equation describing the motion of the paper cup. Level N | 11  Physics N c. Solve the differential equation showing how the speed of the paper cup varies with time. d. Draw the velocity-time graph (v-t) in the coordinate system below. Label the intercepts and the asymptotes in terms of the given. Label the graph v1. On the same coordinate system, show the variation of the velocity of another identical paper cup that has been released from the same height with zero initial speed. Label this graph v2. e. The paper cup reaches its terminal velocity in a shorter period than that predicted by the model. Suggest a reason to explain this observation. Level N | 12  Physics N Chapter 6 Work and Kinetic Energy Section 6.1 Work (43) 1. A force of magnitude 20.0 N acts on a box at an angle of 60 with the horizontal, as shown below. The box then moves horizontally through a distance of 10.0 m. What is the work done by F? (44)2. A body of mass 5.00 kg moves along a horizontal rough surface. The body is subject to a force of magnitude 50.0 N at an angle of 30.0 with the horizontal. If the coefficient of kinetic friction is 0.150, what is the net work done on the body moving a distance of 100 m? (45)3. [G] Marc exerts a constant force of magnitude 150 N to move a crate over a distance of 5.0 m. The crate is lower than his torso, so he must push at an angle of 60° to the direction of motion. What is the amount of work done by Marc on the crate during this displacement? (46) 4. [G] Describe when the work done by a force is positive, negative, or zero. Section 6.2 Kinetic Energy and the Work-Energy Theorem (47) 5. A box of mass 4.0 kg, initially at rest, is acted upon by a net horizontal force of magnitude 8.0 N. What is the final speed of the box after moving a distance of 4.0 m? (48) 6. A package of mass 8.00 kg is pushed along a straight line across a smooth horizontal floor by means of a constant horizontal force of magnitude 16.0 N. The package has a speed of 3.00 m/s when it passes through point A and a speed of 5.00 m/s when it reaches point B. What is the distance AB? (49)7. A crate of mass 60.0 kg is released from rest on a rough inclined plane making an angle  = sin−1(2/7) with the horizontal. After traveling a distance of 90.0 m, the crate reaches a speed of 12.0 m/s. Calculate the coefficient of kinetic friction between the box and the ground. Level N | 13  Physics N (50)8. [G] The speed of a free-falling stone with mass 2.0 kg changes from 3.0 m/s to 6.0 m/s over a certain segment of its path. Find the work done by gravity along the mentioned path. Section 6.3 Work and Energy with Varying Forces (51)9. A spring of spring constant k obeys Hooke’s law, F = −kx, where F is the variable force exerted by the spring and x the displacement of its free end. Calculate the work done by the spring on a particle attached to it and that extends the spring from its natural length to an extension a, as shown below. (52)10. Two forces F1 = 3xi (N) and F2 = 4x2i (N) act on the same particle displacing it from position r1 = (20 cm)i to position r2 = (10 cm)i. What is the net work done on the particle? (53)11. A 0.500 kg object moves on a horizontal circular track with a radius of 2.50 m. An external force of magnitude 3.00 N, always tangent to the track, causes the object to speed up as it goes around. If the object starts from rest, what would its speed be at the end of one revolution? (54) 12. [G] A force F = e−3x (N), where x is in meters, is used to move a box between x = 1.0 m and x = 5.0 m. What is the work done by the force over the given distance? Level N | 14  Physics N (55)13. [G] Find the work done by an elastic force F used to stretch a spring from 8 cm to 16 cm. A graph of the dependence of the elastic force on the length of the spring is shown in the figure below. (56)14. [G] What is the work needed to stretch a spring, having a force constant equal to 40 kN/m, by 0.5 cm from equilibrium? Section 6.4 Power 15. A jet plane produces a thrust of 1.50 × 104 N when it is traveling at 300 m/s. (57) What is the instantaneous power of the plane? (58) 16. A man of mass 80.0 kg climbs up 25 stairs, each of height 20.0 cm, in 20.0 s. What is the average power generated by the man? Take g = 10.0 m/s2. (59) 17. [G] Using a force of 80.0 N, a man raises a bucket of water steadily from a well 10.0 m deep in 20.0 s. What power does this man generate? (60)18. [G] What power does a scooter engine generate at a speed of 36 km/h with a traction force of 250 N? Level N | 15  Physics N (61) 19. [T] Two blocks, of masses M and M/2, are connected by a long string passing over a light frictionless pulley. The apparatus is released from rest. a. Derive an expression for the speed of the block of mass M in terms of the distance d it descends. b. Now the blocks-pulley system is replaced by a uniform rope of length L and mass M. Initially, the rope is hung around the midpoint, such that one of the free hanging parts is only slightly longer than the other. The rope is then released from rest, and at some time later the difference between the lengths of ropes is y, as shown below. Express your answers to parts b, c, and d in terms of y, L, M, and fundamental constants. i. Determine an expression for the force of gravity on the right hanging part of the rope as a function of y. ii. Determine an expression for the force of gravity on the left hanging part of the rope as a function of y. c. Derive an expression for the work done by gravity on the rope as a function of y, assuming y is initially zero. d. Derive an expression for the speed u of the rope as a function of y. Level N | 16  Physics N e. In this part, the two previous cases will be considered separately. The hanging block of mass M and the right end of the rope are each allowed to fall a distance L/2. The string is long enough that the block of mass M/2 does not hit the pulley. Indicate whether v from part a or u from part d is greater after the block and the end of the rope have traveled the same distance L/2. _____v is greater _____u is greater _____the speeds are equal Justify your answer. Level N | 17  Physics N Chapter 7 Potential Energy and Energy Conservation Section 7.1 Gravitational Potential Energy (62) 1. A box of mass 0.200 kg is shot horizontally with an initial speed v0 = 6.00 m/s along a rough track that meets a smooth inclined plane at an angle of 30 with the horizontal. The coefficient of kinetic friction of the horizontal plane is 0.200 and its length is 5.00 m. a. With what speed does the box reach point O? b. What is the distance traveled by the box along the inclined plane before it stops and moves back down? (63) 2. How are the work done by gravity and the change in the gravitational potential energy related? Explain by giving an example. (64)3. A cart rolls along a horizontal track with a speed of 10.0 m/s, and then rolls up a hill. Neglecting all frictional forces, what height will the cart stop at? Take g = 10.0 m/s2. (65)4. The seat of a swing is 0.50 m above the ground when it is stationary. A girl swings and passes through the lowest point with a speed of 6.2 m/s. What is the height of the seat, above ground level, when the girl first comes to rest? Take g = 9.80 m/s2. (66)5. A skateboarder moves from rest down a curved frictionless ramp in the shape of a quarter circle of radius 3.00 m. The skater and the skateboard have a total mass of 70.0 kg. Take g = 10.0 m/s2. a. What is the speed of the skateboarder at the bottom of the ramp? Level N | 18  Physics N b. In real life, the ramp is not frictionless and the speed of the skater at the bottom of the ramp is 6.00 m/s instead of the value found in part a. What is the work done by friction on the skater? (67) 6. [G] A body of mass 1.00 kg has a potential energy of 10.0 J. How high is the body above the ground if the zero reading of gravitational potential energy is at ground level? (68) 7. [G] A stone is thrown vertically upward from the ground level at a velocity of 10.0 m/s. At what height is the kinetic energy of the stone equal to its gravitational potential energy? Take g = 10.0 m/s2. (69)8. [G] A 2.0 kg cart is given an initial speed of 5.0 m/s at the bottom of a rough ramp making an angle of 20° with the horizontal. A constant frictional force of 10.0 N acts on the cart during its motion. What is the maximum distance d traveled by the cart along the ramp? Section 7.2 Elastic Potential Energy (70) 9. A block of mass 1.50 kg is forced against a horizontal massless spring of spring constant 200 N/m, compressing it a distance of 0.150 m. When the block is released, it slides along a horizontal frictionless floor. Take g = 10.0 m/s2. a. What is the speed of the block at the moment it is released from the spring? b. If the block slides along a rough horizontal surface where the coefficient of kinetic friction between the table and the block is 0.300, how far does the block move before it stops? (71)10. The 2.0 kg block shown in the diagram below slides from rest down the frictionless chute. The radius of the ramp (quarter circle) is 2.0 m. Take g = 10.0 m/s2. a. What is the velocity of the block as it reaches the spring bumper at the end? b. What will the maximum compression of the spring bumper be knowing that the spring constant is 200 N/m and that all the kinetic energy is transformed into elastic potential energy? Level N | 19  Physics N (72)11. [G] A force of 250 N keeps a certain spring stretched by 50 cm. What is the elastic potential energy stored in the spring? Section 7.3 Conservative and Non-conservative Forces (73) 12. What are some properties of the work done by a conservative force? Give an example of a conservative force. (74)13. A particle situated at the origin of an xy-plane, having unit vectors i and j, is acted upon by a force F = Cxj, where C is a constant. The particle moves in the counterclockwise direction around the square loop and gets back to its initial position. a. Determine the work done by the force on the particle along each leg of the square. b. Is the force acting on the particle conservative or non-conservative? Explain. c. Can the force be represented by a potential energy function U? Explain. (75) 14. [G] List the properties of work done by a conservative force. (76)15. [G] What is the general form of the law of conservation of energy and how is internal energy related to the work done by non-conservative forces? Section 7.4 Force and Potential Energy (77)16. The potential energy of a body is given by: U(x) = −4x3 + 3x2 + 1, where U is in joules and x in meters. a. What is the force F(x) acting on the body? b. Determine the positions where the body is at equilibrium. Level N | 20  Physics N Section 7.5 Energy Diagrams (78)17. Indicate, for the potential-energy function below, the abscissa of the points of stable equilibrium and unstable equilibrium. Justify your answer. (79)18. [T] A 1.5 kg particle is moving along the x-axis in a region where its potential energy as a function of x is given by U(x) = 0.25x2, where U is expressed in joules and x in meters. When the particle passes through point x = 4.0 m, its velocity is –2.0 m/s. All the forces acting on the particle are considered to be conservative. a. Calculate the total mechanical energy of the particle. b. Calculate the x-coordinates of the points at which the particle has half its maximum kinetic energy. c. Calculate the maximum value of the particle's speed. d. Calculate the maximum value of the particle's acceleration. e. On the axes below, sketch graphs of the object’s position x versus time t and kinetic energy KE versus time t. Assume that at t = 0, the particle reaches its greatest speed, moving in the positive direction. The two graphs should cover the same time interval and use the same scale on the horizontal axes. Level N | 21  Physics N (80) 19. [T] A box of mass m is released from rest at point A, as shown in the figure below, and moves along a track to point E. The box falls freely between points A and B, which are a distance of R/2 apart. Then, it moves along the circular arc of radius R between points B and D. Assume the track is frictionless from point A to point D. a. On the dot below that represents the box, draw and label the forces that act on the box when it is at point C, which is at an angle θ from the vertical through point D. b. Determine, in terms of θ and the magnitudes of the forces drawn in part a, an expression for the magnitude of the centripetal force acting on the box at point C. c. Derive an expression for the speed uD of the box as it reaches point D in terms of M, R, and fundamental constants. d. A force acts on the box between points D and E and brings it to rest at point E. If the box is brought to rest by friction, calculate the numerical value of the coefficient of friction μ between the box and the track. Level N | 22  Physics N e. Now, consider the case in which there is no friction between the box and the track, but instead the box is brought to rest by a braking force expressed as –kv2, where k is a constant and v is the velocity of the box. Express all algebraic answers to the following in terms of m, R, k, and fundamental constants. i. Derive, but do not solve, the differential equation for v(t). ii. Solve the differential equation you derived in part i. iii. On the axes below, sketch a graph of the magnitude of the acceleration of the box as a function of time. On the axes, explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate. (81) 20. [T] Students are to conduct an experiment to investigate the relationship between the terminal speed of a sphere and its radius. They take steel spheres of density ρ and different radii r and drop them into a tall glass filled with glycerin. They then measure the time t necessary for each ball to cover the distance h = 60 cm. The students’ data is given in the table below. Radius r (mm) Time (s) Height (cm) uT (m/s) 2.0 8.1 60 2.5 5.2 60 3.0 3.6 60 3.5 2.6 60 Students learned from their textbook that the magnitude of the drag force FD on a sphere moving in a liquid is given by FD = 6rv , where  is a constant (usually referred to as viscosity). Use g = 9.8 m/s2. a. Using this relationship, derive an expression relating the terminal speed uT to the radius of the ball r. b. i. Assuming the functional relationship for the drag force above, use the grid below to plot a linear graph as a function of r to verify the relationship. Use the empty boxes in the data table, as appropriate, to record any calculated values you are graphing. Label the vertical axis as appropriate, place numbers on both axes, and draw the best straight-line fit to the points. Level N | 23  Physics N ii. Given that the density of steel is 8.05 g/cm3, calculate the viscosity of glycerin. c. On the same pair of axes below sketch i. the kinetic energy versus time from the time the steel ball is released up to the time t = T that the ball has fallen the distance h. Label the graph K. ii. the potential energy versus time from the time the steel ball is released up to the time t = T that the ball has fallen the distance h. Take the bottom of the glass as a zero potential energy level. Label the graph U. Level N | 24  Physics N Chapter 8 Momentum, Impulse, and Collisions Section 8.1 Momentum and Impulse (82) 1. A baseball of mass 150 g is thrown horizontally against a wall at 20.0 m/s and rebounds (also horizontally) at 15.0 m/s. a. What is the impulse of the net force acting on the ball during collision? b. The ball is in contact with the wall for 15.0 ms. Calculate the magnitude of the average force that the wall exerts on the ball. (83)2. A ball of mass 0.50 kg hits a wall horizontally at 25 m/s and rebounds horizontally at 18 m/s. What is the momentum of the ball right before the collision? What is its momentum after it rebounds off the wall? (84) 3. The graph below shows how a force exerted on a certain body varies with time. What is the impulse given to that body during 3.0 s? (85)4. A particle of mass 6.0 kg moves with a velocity of (3.0 m/s)i − (2.0 m/s)j. After 3.0 s, the particle has a velocity of (7.0 m/s)i + (3.0 m/s)j. What are the magnitude and direction of the constant force acting on the particle? (86) 5. [G] A car of mass 1,200 kg is moving at a speed of 72 km/h. Find its momentum. (87)6. [G] A body of mass 1.0 kg moves with a velocity 2.0 m/s along the Ox axis. A force of 4.0 N acts along the direction of motion for 2.0 s. Determine the final speed of the body. (88)7. [G] A 5.0 kg ball moving at a speed of 60 m/s hits a wall and bounces off it with the same speed. What will the impulse of the force received by the wall be if the ball flies and bounces perpendicular to it? Level N | 25  Physics N (89)8. [G] A 0.40 kg ball moving at a speed of 12 m/s hits a vertical wall at a right angle. What is the average force acting on the ball if it rebounds at the same speed and the interaction with the wall lasts for 10 ms? (90)9. [G] A particle of mass 3.0 kg is initially moving along the positive x-axis. At t = 0, a force F is applied along the direction of motion of the particle for 2.0 s. The force obeys the equation F = 6t ‒ 3t2. What is the impulse of the force in the time interval [0, 2.0 s]? Section 8.2 Conservation of Momentum (91)10. A bullet of mass m is fired by a rifle of mass M. Initially, both the bullet and the rifle were at rest. Express the velocity of the rifle after the bullet was shot in terms of the bullet’s speed. In what direction does the rifle move after the bullet is shot? (92)11. A car of mass 800 kg moving at 25.0 m/s collides with a small truck of mass 1200 kg moving in the same direction at 10.0 m/s. After the collision, the two vehicles combine and move with the same velocity. What is the velocity of the system after the collision? (93)12. [G] A 70.0 kg ice skater, standing stationary in an ice rink, throws a 3.00 kg stone in a horizontal direction at a speed of 8.00 m/s. What will be the speed of the ice skater after throwing the stone? Take g = 10.0 m/s2. Section 8.3 Conservation of Momentum (94)13. a. Is the collision studied in question 2 in Section 8.2 elastic or inelastic? b. Calculate the energy lost by the system after the collision. (95)14. At a road intersection, a red car of mass 950 kg is moving eastward at a speed of 20.0 m/s. A blue pickup truck, of mass 1750 kg traveling north at 25.0 m/s, runs a red light and collides with the car. As a result, the two vehicles stick together and continue moving as one. Level N | 26  Physics N a. What is the momentum of each vehicle before the collision? b. What is the velocity (magnitude and direction) of the two vehicles after the collision? c. Show that the collision is inelastic. d. In what form is the energy dissipated after the collision? (96) 15. [G] Differentiate between elastic and inelastic collisions. Section 8.4 Elastic Collisions (97)16. A bowling ball of mass 6.80 kg is moving along a bowling alley with a speed of 8.00 m/s. The ball collides elastically with a stationary bowling pin of mass 1.50 kg. After the collision, the bowling ball deflects to the left of its original path of motion at an angle of 12.0 and its speed reduces to 5.80 m/s, as shown in the figure below. On the other hand, the pin is deflected to the right of the original path of motion at a certain angle . y 8.00 m/s 12.0 x  Top view v’pin a. Assuming that the collision is elastic, calculate the speed of the bowling pin after the collision. b. At what angle does the pin deflect? (98)17. [G] A ball of mass m1 = 200 g, moving at a speed of 10 m/s, collides with a stationary ball with a mass m2 = 800 g. The collision is perfectly elastic. What will the speeds of the balls be after collision? Section 8.5 Center of Mass (99)18. Three particles forming a system have the following properties: P1 is of mass 2.0 kg and located at −i + 2j (m), P2 of mass 1.0 kg and located at 3i + j (m), and P3 of mass 2.0 kg and located at 2i − 2j (m). What is the center of mass of the system? (100) 19. When does the center of mass of a system of particles have a constant velocity? Level N | 27  Physics N (101)20. [G] A boat floats motionlessly in a lake. Two fishermen are sitting at the stern and on the bow of the boat 4.0 m apart. The mass of the boat is 200 kg, while the masses of the fishermen are 85 kg and 75 kg. Assume that the center of mass of the boat is at a distance of 2.0 m from its stern at the height of the top of the boat. Find the position of the center of mass of the given system of bodies. (102) 21. [T] A box of mass m is to be pulled up a ramp by a rope, as shown below. The magnitude of the acceleration of the box as a function of time t can be modeled by the equations.  t  T a = amax cos T , t  0; 2     a = 0, t  T  2 where amax and T are positive constants. The hill is inclined at an angle θ above the horizontal, and friction between the box and the ramp is negligible. Express your answers in terms of the given quantities and fundamental constants. a. Derive an expression for the velocity of the box as a function of time during the accelerating phase. Assume the box starts from rest. b. Derive an expression for the work done by the net force on the box from rest until terminal speed is reached. c. Determine the magnitude of the force exerted by the rope on the box at terminal speed. d. Derive an expression for the total impulse given to the box during the accelerating phase. e. Now, suppose that—although the initial acceleration is still amax—the magnitude of the acceleration was modeled as decreasing linearly with time. On the axes below, sketch the graphs of the force exerted by the rope on the box for the two models from t = 0 to t > T/2. Label the original model F1 and the new model F2. Level N | 28  Physics N (103)22. [T] A small block of mass m = 0.50 kg is placed on a large triangular slab of mass M = 3.0 kg, as shown below. The height of the triangular slab, whose inclination is α = 37, is 3.0 m. Initially, both the block and the slab are at rest. There is no friction between the slab and the horizontal surface, as well as between the block and the slab. Use g = 10 m/s2. a. On the dots below that represent the block and the slab, draw and label vectors to represent the forces acting on each as the block slides on the slab. b. Calculate the acceleration of the i. slab relative to the ground. ii. box relative to the slab. c. Let v be the speed of the box as it leaves the slab and V the speed of the slab at that moment. Express the relation between v and V in terms of m, M, and α. Level N | 29  Physics N d. How would the value of V change if there was friction between the block and the slab? ___V would increase. ___V would decrease. ___V would remain the same. Justify your answer. (104)23. [T] A ball of mass 200 g is released from rest at a fixed height of 1.80 m above the ground. When it hits the ground, a force sensor measures the force exerted by the ground on the ball. The data obtained from the sensor is shown below. Use g = 10 m/s2. Time (s) Force (N) 0 0 0.05 0.1 0.1 5.2 0.15 10.3 0.175 13 0.2 10.4 0.25 5.6 0.3 0.2 0.33 0 a. On the graph paper below, plot the force-time graph. Label the axes and sketch a smooth curve through the points. b. Determine the impulse imparted by the ground on the ball. c. Calculate the speed of the ball when it leaves the ground. Level N | 30  Physics N d. Determine the magnitude of work done on the ball by dissipative forces during the impact. Level N | 31  Physics N Chapter 9 Rotation of Rigid Bodies Section 9.1 Angular Velocity and Acceleration (105)1. A wheel has an angular position that varies with time as follows:  = (6.0 rad/s2)t2. a. How many revolutions does the wheel make in the first 5 seconds of rotation? b. What is the angular velocity of the wheel at t = 2.0 s? c. What is the angular acceleration of the wheel at t = 2.0 s? (106) 2. A wheel is turning with a constant angular speed of 3.0 rad/s. What is the time taken by the wheel to complete one full revolution? (107)3. The Ferris wheel at a local amusement park takes 40 s to complete one revolution. What is the average angular speed of the wheel? (108) 4. A car is moving along a horizontal road. Its tires are all rotating with an angular velocity of 53.5 rad/s. The driver accelerates uniformly for a duration of 4.50 s. As a result, each tire reaches an angular velocity of 58.0 rad/s. What is the average angular acceleration of a tire during the 4.50 s? (109)5. [G] The linear velocity of the rim points of a uniformly rotating disk is 3.0 m/s, and that of the points located 10 cm closer to the axis of rotation is 2.0 m/s. What is the angular velocity of the disk? (110) 6. [G] The rotational motion of a wheel is governed by the expression  = 1 + 2t + t 3 (rad). Find the instantaneous angular velocity at t = 2.0 s of a point lying on the wheel’s rim. What is the direction of rotation? (111) 7. [G] The rotational motion of a wheel is governed by the expression θ = 1+ 2t + t3 (rad). Find the instantaneous angular acceleration at t = 2.0 s of a point lying on the wheel’s rim. What is the direction of rotation? Section 9.2 Rotation with Constant Angular Acceleration (112)8. Initially, a grinding wheel has an angular velocity of 24.0 rad/s. The wheel accelerates at 30.0 rad/s2 for two seconds only. a. What is the angular velocity of the wheel after 2.00 s? b. Through what angle did the wheel turn during the two seconds? c. The wheel then starts decelerating at a constant rate until it finally stops after turning through 432 rad. i. What is the duration of the decelerating phase? Level N | 32  Physics N ii. What was the wheel’s angular acceleration during this phase? (113) 9. When a DVD player is turned off, a disc inside it, initially rotating counterclockwise at 4.60 × 103 rev/min, stops rotating. The disc is assumed to decelerate uniformly at a rate of −50.0 rad/s2. When the DVD player is off, a red dot is marked on the disc, as shown below. a. What is the magnitude of the disc’s angular velocity 3.00 s after the DVD player was turned off? b. What is the angular displacement of the red dot 3.00 s after the DVD player was turned off? c. What is the time needed for the disc to stop? d. How many revolutions would the red dot make before it stops? 10. [G] At t = 0, a wheel rotating with an angular velocity ω = 5.0 rad/s starts (114) slowing down uniformly until it stops. If the angular acceleration is α = −2.0 rad/s2, how many degrees will the wheel turn in the first 2.0 s and what is, then, the wheel’s angular velocity? Section 9.3 Relating Linear and Angular Kinematics (115) 11. A girl is playing on a merry-go-round of diameter 3.0 m. At some instant, the magnitude of her angular acceleration is 0.50 rad/s2 and that of her centripetal acceleration is 3.0 m/s2. If the girl is at the edge of the merry-go-round, what is the magnitude and direction of her net acceleration at that instant? (116)12. A carousel in an amusement park starts rotating from rest with a constant angular acceleration at t0 = 0 till t1 = 1.00 min. During this time interval, the carousel completes 5 turns. A pink horse and a yellow horse are 2.00 m and 3.00 m away from the center of the carousel, respectively. a. What is the angular displacement of the pink horse between t0 and t1 if it has described an arc length of 4.00 m? b. What are the values of the angular acceleration and tangential acceleration of the yellow horse during the one-minute interval? Level N | 33  Physics N c. What is the speed of the yellow horse at t1? Is this speed equal to that of the pink horse? Explain without performing any calculations. d. What is the magnitude of the centripetal acceleration of the yellow horse at t1? e. Determine the characteristics of the net acceleration of the yellow horse at t1. 13. [G] A wheel at rest starts rotating with an angular acceleration α = 2.0 rad/s2. (117) After 0.5 s, the magnitude of the wheel’s acceleration is 13.6 cm/s2. Find the radius of the wheel. Section 9.4 Energy in Rotational Motion (118) 14. Four identical particles, each of mass 2.0 kg, are arranged in an x-y plane, as shown below. The particles are connected by light sticks to form a rigid body. If a = 1.0 m, what is the moment of inertia of this array about the y-axis? (119)15. A uniform disc of mass 2.5 kg and radius 65 cm is initially at rest. The disc is free to rotate about a horizontal axis that passes perpendicularly through its center. A force F is applied tangentially to the disc for 5.0 seconds, resulting in an increase in the disc’s angular speed to 12 rad/s. a. Calculate the final kinetic energy of the disc. b. What is the disc’s angular acceleration? (120)16. A uniform basketball has a mass of 300.0 g and a radius of 15.0 cm. The basketball is released with zero initial speed from a height of 1.50 m on a rough ramp, as shown below. The ball rolls without slipping. Take the horizontal plane passing through the bottom of the ramp as a reference for gravitational potential energy and g = 10.0 m/s2. By applying the principle of conservation of energy, what is the translational speed of the basketball at the bottom of the ram? Level N | 34  Physics N (121) 17. [G] Four small spheres, each of mass 0.30 kg, are arranged in the shape of a square of side 0.50 m. The spheres are connected by extremely light rods. What is the moment of inertia of the system about an axis through one of the spheres and perpendicular to its plane? (122) 18. [G] A solid disk of mass 5.0 kg rotates in its place at a constant speed of 2.0 m/s. Find the kinetic energy of the disk. Section 9.5 Parallel-Axis Theorem (123)19. Using the parallel-axis theorem, derive the expression of the moment of inertia of a thin rod of length l and mass M whose axis of rotation passes through one of its ends. (124) 20. [G] Determine the moment of inertia of a thin uniform rod relative to an axis perpendicular to the rod and passing through a point located 1/4 of its length from the end of the rod. The length of the rod is 0.50 m and its mass is 0.20 kg. Section 9.6 Moment-of-Inertia Calculations (125)21. A hollow cylinder, shown below, has a density , an inner radius R1, an outer radius R2, and a height h. a. Calculate the moment of inertia of the cylinder about the axis of rotation that passes through its center of mass. b. Deduce the moment of inertia of a solid cylinder and a thin cylindrical shell. Level N | 35  Physics N (126)22. [T] A hoop of mass 0.50 kg is released from rest from the top of a slanted roof that is 2.0 m long and angled at 37 with the horizontal, as shown below. The hoop rolls along the roof without slipping. The moment of inertia of a hoop of mass M and radius R about its center of mass is MR2. Use g = 10 m/s2. a. On the figure below, draw and label the forces (not components) acting on the hoop at their points of application as the loop rolls along the roof. b. Calculate the force due to friction acting on the hoop as it rolls along the roof. If you need to draw anything, other than what you have drawn in part a, to assist in your solution, use the space below. Do not add anything to the figure in part a. c. Calculate the linear speed of the center of mass of the hoop, vcom, when the hoop reaches the bottom edge of the roof. d. A wagon containing a box is at rest on the ground below the roof, in such a way that the hoop falls a vertical distance of 1.5 m, then lands and sticks in the center of the box. The total mass of the wagon and the box is 2.5 kg. Calculate the horizontal speed of the wagon immediately after the hoop lands in it. Level N | 36  Physics N Chapter 10 Dynamics of Rotational Motion Section 10.1 Torque (127) 1. The diagram below shows two forces F1 and F2 acting on a rod. Determine the signs of the torques due to F1 and F2. (128)2. A rod is pivoted about its center. Two forces, each of magnitude 5.0 N, act at 4.0 m and 2.0 m from the pivot point, as shown below. What is the net torque about the pivot? (129) 3. The diagram below shows three forces applied at points A, B, and C of a uniform beam pivoted at end O. At which of the points is the greatest torque with respect to O produced? (130) 4. [G] What three factors affect the torque created by a force relative to a specific pivot point? How is the direction of the torque vector determined? Level N | 37  Physics N Section 10.2 Torque and Angular Acceleration for a Rigid Body (131) 5. A uniform spherical ball, of mass 300 g and diameter 0.50 m, is resting on a horizontal surface in a vertical plane. A force F, of magnitude 6.2 N, is applied on the ball tangent to one of its extremities. a. Neglecting frictional forces, what is the resultant net torque acting on the ball about a diameter of the ball perpendicular to the plane of the figure? b. Deduce the angular acceleration of the ball. (132)6. A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis passing through their respective centers. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown below. Rank, from least to greatest, the angular accelerations of the three objects. (133) 7. [G] A mass of 6.0 kg is tied to a cord wound around a disk of radius 0.40 m. Find the moment of inertia of the disk knowing that the load drops with an acceleration of 4.0 m/s2. Use g = 10 m/s2. Section 10.3 Rigid-Body Rotation About a Moving Axis (134)8. A solid sphere, of mass m, released from rest from a height h rolls down an inclined plane. The sphere then rolls around a loop of radius R, as shown below. a. Based on the principle of conservation of mechanical energy, derive an expression of the sphere’s speed at the top of the loop. b. What should the minimum height h be from which the sphere has to be released for it not to fall off the loop? Assume that the radius of the sphere is much smaller than that of the loop. Level N | 38  Physics N 9. Consider an Atwood’s machine made of two blocks of different masses m1 and (135) m2 attached by a string around a disk of radius R and mass M, as shown below. Given m1 = 5.00 kg, m2 = 7.00 kg, M = 1.00 kg, R = 4.00 cm, and g = 9.80 m/s2. a. By applying Newton’s second law for translational and rotational motion, derive an expression for the acceleration of the system. Deduce the value of the system’s acceleration. b. What are the magnitudes of the tensions T1 and T2 in each section of the rope? c. Determine the net torque acting on the disk and its angular acceleration. (136) 10. [G] A solid disk of mass 5.0 kg rolls without sliding along a horizontal plane with a speed of 2.0 m/s. Find the kinetic energy of the disk. (137)11. [G] A wheel with a radius of 0.50 m and a mass of 4.0 kg rolls without friction along an inclined plane 4.0 m long that is inclined at an angle of 30°. Determine the moment of inertia of the wheel if the speed of its center of mass at the end of the movement is 5.0 m/s. The acceleration of gravity is assumed to be 10 m/s2. Section 10.4 Work and Power in Rotational Motion 12. A bolt is tightened with a force of torque 5.50 × 10−2 N.m through 25 turns. (138) What is the total work required to tighten the bolt? Level N | 39  Physics N (139)13. A disk, of mass 1.50 kg and radius 20.0 cm, is acted upon by a constant torque that increases its angular speed from 0 to 15.0 rad/s. What is the total work done by the constant torque? (140) 14. A 5.00 kg grinding wheel is in the form of a solid cylinder of radius 0.80 m. A constant torque rotates it through 4.0 complete turns in 10.0 seconds. a. What is the work done by the torque? b. What is the power due to the torque? (141)15. [G] A flywheel in the form of a solid disk, with I = 2.0 kg.m2, decelerates from 8.0 rev/s to 0 in one minute. Determine the constant torque and the work done by the braking force. (142)16. [G] A string wrapped around a pulley is pulled with a constant downward force of magnitude 50 N. The radius of the pulley is R = 0.10 m and its moment of inertia is I = 2.5×10−3 kg.m2. If the string does not slip, what is the angular velocity of the pulley after 1.0 m of the string has unwound? Assume the pulley starts from rest. (143)17. [G] Find the power due to a torque of 20 N.m acting on a rigid body if it rotates with an angular velocity of 20 rad/s. Section 10.5 Angular Momentum (144)18. A point-like particle of mass 125 g is rotating in a circular path of diameter 1.40 m with a speed of 23.0 m/s. What is the angular momentum of the particle with respect to the axis passing perpendicularly through the center of the circle? (145)19. A system consists of a uniform rod of moment of inertia Irod = 60.00 kg.m2 and a uniform sphere of moment of inertia Isphere = 2.100 kg.m2, as shown below. The system rotates about  at a constant angular speed of 25.00 rad/s. What is the angular momentum of the system? Level N | 40  Physics N (146)20. A hollow thin-walled sphere of mass 12.0 kg and diameter 50.0 cm is rotating about an axis that passes through its center. The angular velocity of the sphere is given by z = (20.0 rad/s3)t2 + (5.00 rad/s2)t. a. Express the angular momentum of the hollow sphere in terms of t and find its value at t = 5.00 s. b. What is the expression of the net torque on the sphere? Find its magnitude at t = 5.00 s. (147)21. [G] A proton spiraling around a magnetic field undergoes circular motion in the plane of a paper, as shown below. The circular path has a radius of 0.40 m and the proton has a velocity of 4.0 × 106 m/s. Given the mass of a proton = 1.67 × 10‒27 kg. a. What is the angular momentum of the proton about the origin? b. What is the direction of the angular momentum? (148) 22. [G] A thin uniform rod with a length of 0.50 m and a mass of 0.40 kg rotates with an angular velocity of 30 rad/s around an axis passing perpendicularly to the rod through its middle. Determine the angular momentum of the rod. Section 10.6 Conservation of Angular Momentum (149)23. A 42.0 kg boy is spinning on a merry-go-round of 625 kg mass and 2.50 m radius in a horizontal plane. The boy walks from the edge towards the center of the merry-go-round. Initially, the merry-go-round was spinning with an angular speed of 0.40 rad/s. Neglect the effect of frictional forces. What is the angular speed of the system when the boy is 1.00 m away from the center? (150)24. A uniform disk of moment of inertia I1 = 2.5 kg.m2 is rotating clockwise in a horizontal plane with an angular speed of 15 rad/s. A second uniform disk of moment of inertia I2 = 1.3 kg.m2, initially at rest, falls on the rotating disk such that their centers coincide. The system formed by the two disks starts rotating at an angular speed . Neglect the effect of frictional forces. Level N | 41  Physics N a. Is the angular momentum of the system conserved? If yes, determine its value. b. What is the value of the angular speed ? c. What is the loss of energy as a result of the second disk’s fall? (151)25. [G] A horizontal platform with a mass of 160 kg rotates around a vertical axis passing through the center of the platform with a frequency of 0.25 rev/s. A man weighing 80 kg is standing on the edge of the platform. What frequency does the platform begin to rotate with if the man moves from the edge of the platform to its center? Consider the platform as a homogeneous disk and the man as a point mass. (152)26. [T] A system consists of a bullet of mass m and a uniform rod of mass M and length d. The rod is suspended by a frictionless pivot and is initially vertical. The Md 2 moment of inertia of the rod about the pivot is I =.The bullet, traveling 3 horizontally, collides with the free end of the rod and gets embedded in it. As a result of this collision, the rod rotates through an angle α. Express all answers in terms of m, M, α, d, and fundamental constants. a. Derive an expression for the initial speed u of the bullet. Now, assume that M >> m. b. Simplify the answer you obtained in part a using the aforementioned assumption. c. Derive an expression for the change in the momentum of the bullet-rod system during the collision. d. Explain why the momentum of the bullet-rod system is not conserved in the collision. e. Assume now that the collision between the bullet and the rod was perfectly elastic. How would that affect the angle θ through which the rod rotates? __ α > θ __ α = θ __ α < θ Justify your answer. Level N | 42  Physics N (153)27. [T] An Atwood machine is made up of two blocks, of masses m and 3m, and a solid disk of radius R and mass m. The machine is placed on an inclined plane (angle of elevation α). There is no friction between the block, the disk, and the inclined plane. Assume that the cord does not slip on the disk. The moment of inertia of a disk of mass 1 m and radius r is given by I = mr 2. 2 a. On the diagram below, show the forces acting on blocks m and 3m. Express your answers in terms of m, R, α, and fundamental constants. b. Write, but do not solve, a system of algebraic equations used to determine the tensions in the cords and the acceleration of the blocks. c. Determine the acceleration of the block m. Sand gets stuck between the disk and the inclined plane, resulting in friction between the disk and the inclined plane, but not between the blocks and the plane. As a result, the blocks begin moving uniformly along the plane. Assume that the sand is uniformly distributed across the disk and the coefficient of friction between the disk and the inclined surface is μ. d. Determine the torque by friction on the disk, f, in terms of m, R, α, μ, and fundamental constants. e. Assuming α = 37, what is the value of μ? Level N | 43  Physics N (154) 28. [T] A trebuchet is a catapult that uses a long arm to throw a projectile. The trebuchet is made of a uniform beam of mass M and length 3L. It is placed on a fulcrum, located L meters away from the beam's right end. To the right end of the beam, a heavy object of mass 5M is attached. When the trebuchet is released from rest, it turns on the fulcrum, projecting a small mass m, initially at rest in a sling, attached to the trebuchet's left end. The rock leaves the sling when the beam is vertical. The moment of inertia of a uniform beam of mass m and length l, rotating about its 1 2 center, is ml. Assume m a0 ___a = a0 ___a < a0 Justify your answer. d. Determine the speed vl with which the rock is launched. e. Determine the range of the rock. Level N | 44  Physics N (155)29. [T] A barbell consists of two balls, each of mass m and radius 2R, connected by a cylinder of mass m and radius R. The barbell is released from the top of an inclined plane of height h and rolls to the bottom. Assume it does not slip. At the bottom of the inclined hill, the barbell collides with a light non-linear spring and compresses it by l m before spontaneously coming to 2 rest. Given: the moment of inertia of a solid sphere of mass M and radius r is Mr 2 5 and the moment of inertia of a cylinder of mass M and radius r about its central axis is 1 2 Mr. 2 Express all algebraic answers in terms of the given quantities and fundamental constants. a. Determine the moment of inertia of the barbell. b. Determine the speed of the barbell at the bottom of the incline. c. Assume that the force the spring exerts on the barbell can be modeled as F = −kx 2 , where k is an unknown spring constant. i. Derive, in terms of k, an expression for the elastic potential energy of this spring. ii. Express the value for the spring constant k in terms of the given and the fundamental constants. d. If the barbell was rolling with slipping, would you expect the speed of the barbell at the bottom of the incline to be higher or lower? Justify your answer. Level N | 45  Physics N (156)30. [T] The diagram below shows the dimensions of a spool of thread, modeled as three disks of different radii. The rotational inertia of a disk of mass M and radius R 1 about its axis is MR 2. 2 a. Derive an expression for the moment of inertia of a spool of thread rotating about the axis of its symmetry. b. The spool is placed on a rough plane inclined at an angle  with the horizontal and is tied by a thread, as shown below. The spool is initially at rest and the thread is parallel to the inclined plane. Derive an expression for the tension in the thread. c. The thread is cut, and the spool is allowed to roll down the inclined plane without slipping. i. Derive an expression for the linear acceleration of the spool. ii. The length of the inclined plane is L. Derive the expression for the speed of the spool at the bottom of the inclined plane, vb. d. A spool is now placed on an inclined plane with a higher coefficient of static friction and allowed to roll without slipping down the new plane. How would that affect the magnitude of the speed of the spool at the bottom of the inclined plane, v? ___ v > vb ___ v = vb ___ v < vb Justify your answer. Level N | 46  Physics N Chapter 11 Equilibrium and Elasticity Section 11.1 Conditions for Equilibrium (157) 1. What are the two conditions for equilibrium? (158) 2. Is the following body in equilibrium? Section 11.2 Center of Gravity (159) 3. When does a body’s center of gravity coincide with its center of mass? Section 11.3 Solving Rigid-Body Equilibrium Problems (160) 4. A man weighing 900.0 N stands on a uniform horizontal board of length 2.00 m and mass considered to be negligible. The board is located in a vertical plane and fixed on two supports at its extremities, as shown below. Given that the man is 0.50 m away from support 1, what is the magnitude of the reaction exerted by support 1 on the board? (161)5. The figure below shows a uniform rod AB of mass M = 200.0 g pivoted in a vertical plane at point O. The rod is held in equilibrium when a block of mass m = 55.0 g is hanged from side A by a string and a force F is applied perpendicularly to it from side B. The position of the pivot O is such that OB = 2/3AB and OA = 1/3AB. Given: g = 10.0 m/s2. What is the magnitude of F? Level N | 47  Physics N (162) 6. [G] The horizontal beam in the figure below weighs 190 N, and its center of gravity is at its center. Find the tension in the cable. Section 11.4 Stress, Strain, and Elastic Moduli (163) 7. A certain wire stretches 0.90 cm when outward forces of magnitude F are applied to each end. The same forces are applied to a wire of the same material but with three times the diameter and three times the length. By how much would the second wire stretch? (164)8. A 4.00 m long steel beam with a cross-sectional area of 1.00 × 10−2 m2 and a Young’s modulus of 2.00 × 1011 N/m2 is wedged horizontally between two vertical walls. In order to wedge the beam, it is compressed by 0.0200 mm. If the coefficient of static friction between the beam and the walls is 0.700, what is then the maximum mass (including its own) that the beam can bear without slipping? Take g = 10.0 m/s2. (165) 9. A cube, of an exactly 2 cm side, is made of a material with a bulk modulus of 3.5×109 N/m2. When the cube is subjected to a pressure of 3.0 × 105 Pa, what is its volume? Level N | 48  Physics N (166) 10. A shearing force of 50.0 N is applied to an aluminum rod, having a length of 10.0 m, a cross-sectional area of 1.00 × 10−5 m2, and a shear modulus of 2.50 × 1010 N/m2. What is the deformation in the aluminum rod? (167)11. [T] The horizontal uniform plank shown below has a length of 40 cm and a mass of 1.5 kg. The left end of the plank is attached to a vertical support by a frictionless hinge that allows the plank to swing up or down. The midpoint of the plank is supported by a cord that makes an angle of 37 with the rod. A 1.0 kg block is attached to the right end of the rod. Use g = 10 m/s2. a. On the diagram below, draw and label all the forces acting on the plank. Show each force vector as originating at its point of application. b. Calculate the tension in the cord supporting the plank. 1 c. The moment of inertia of a plank about its center of mass is given by ML2 , where 12 M is the mass of the plank and L is its length. Calculate the moment of inertia of the plank-block system about the hinge. d. The cord is now cut and the plank with the weight rotate on the hinge. Determine the speed of the weight just before it hits the vertical stand. Level N | 49  Physics N Chapter 12 Fluid Mechanics Section 12.1 Gases, Liquids, and Density (168) 1. What is the mass of air in a room of dimensions 5.0 m × 5.0 m × 3.0 m? Given air = 1.20 kg/m3. Section 12.2 Pressure in a Fluid 2. An airtight box, with a lid of area 80.0 cm2, is partially evacuated. A force of (169) 600.0 N is required to pull the lid off the box. What is the pressure in the box? Take atmospheric pressure to be 1.01 × 105 Pa. (170)3. A cylindrical tube of cross-sectional area of 4.00 cm2 contains a certain height of acetone. The surface of acetone holds a block of mass 200.0 g. What is the height of acetone if the pressure at point B is 112358 Pa? Given: acetone = 784 kg/m3, atmospheric pressure p0 = 101300 Pa, and g = 9.80 m/s2. (171) 4. A hydraulic press has one piston of diameter 2.0 cm and the other piston of diameter 8.0 cm. What force must be applied to the smaller piston to obtain a force of 1600 N at the larger piston? (172)5. A U tube is filled with a liquid of density 1250 g/L, as shown below. In one of the tube’s vessels, a quantity of oil was poured to a height of 25.0 cm. The elevation of the liquid in the tube is 16.0 cm. Take g = 10.0 m/s2. Level N | 50  Physics N a. What is the pressure of height l of the liquid in the tube? b. What is the density of the oil used? (173) 6. [G] What is the pressure difference between 1.0 m and 2.0 m below the surface of water? (ρwater = 1000 kg/m3 and g = 10 m/s2) Section 12.3 Buoyancy (174) 7. A body weighs 6.5 N in the air. When totally immersed in water, the body weighs 4.0 N. What is the volume of the body? Take g = 10.0 m/s2 and water = 1.0 × 103 kg/m3. (175)8. A wooden boat floats in fresh water with 70% of its volume under water. What is the density of the wood? Take water = 1.0 × 103 kg/m3. (176) 9. [G] State Archimedes’ principle. 10. [G] What force is needed to keep a 50 dm3 granite brick at rest in deep water? (177) Given: ρgranite = 2600 kg/m3, ρwater = 1000 kg/m3, and g = 10 m/s2. Section 12.4 Fluid Flow (178)11. One end of a cylindrical pipe has a radius of 1.5 cm. Water streams steadily out at 7.0 m/s. What is the mass rate at which water is leaving the pipe? Take water = 1.0 × 103 kg/m3. (179)12. Water flows from a 4.0 cm diameter pipe to a 2.0 cm diameter pipe. The speed of flow in the narrow pipe is 8.0 m/s. What is the speed of water in the wide pipe? Section 12.5 Bernoulli’s Equation (180) 13. Water flows through a horizontal pipe. At the wide end of the pipe, water flows at 4.0 m/s. If the difference in pressure between the two ends is 4.5 × 103 Pa, what is the speed of the water at the narrow end? Take water = 1.0 × 103 kg/m3. (181)14. A water line enters a house 2.0 m below ground level. A smaller diameter pipe carries water to a faucet on the second floor that is 5.0 m above the ground. Water flows at 2.0 m/s in the main line and at 7.0 m/s on the second floor. What is the difference in pressure between the main line and the second floor, and which is at the higher pressure? Take water = 1.0 × 103 kg/m3. Level N | 51  Physics N Chapter 13 Gravitation Section 13.1 Newton’s Law of Gravitation 1. The distance between Earth and Venus ranges from 3.80 × 1010 m to 26.1 × (182) 10 m. The masses of Earth and Venus are, respectively, 6.00 × 1024 kg and 4.90 × 10 1024 kg. Treating Earth and Venus as point particles, what is the minimum and maximum gravitational force between the two planets? Take G = 6.67 × 10−11 N.m2/kg2. (183)2. Consider a system of three balls, each of mass 2.0 kg, placed at the vertices of a right-angled triangle, as shown below. Find the magnitude and direction of the net force on the upper ball due to its gravitational interaction with the other two. Take G = 6.67 × 10−11 N.m2/kg2. (184)3. [G] At what height above the surface of Earth is gravity two times less than that on the surface of Earth? The radius of Earth is 6,371 km. Section 13.2 Weight 4. The sun has an approximate mass of 2.0 × 1030 kg and a radius of 7.0 × 105 (185) km. What is the acceleration due to gravity near the sun’s surface? Take G = 6.67 × 10−11 N.m2/kg2. 5. Calculate the acceleration of free fall on the Moon’s surface knowing that the (186) Moon’s radius is about four times smaller than that of Earth and that it is 81 times less massive. (187)6. [G] What is the expression of the gravitational acceleration near the surface of a planet? Section 13.3 Gravitational Potential Energy (188)7. A satellite of mass 1.0 × 103 kg is in orbit around Earth at an altitude of 500 km. We treat the satellite as a point particle. What is the gravitational potential energy of the satellite? Level N | 52  Physics N Given ME = 6.0 × 1024 kg, RE = 6.4 × 106 m, and G = 6.67 × 10−11 N.m2/kg2. (189) 8. What is the minimum speed required of a particle for it to escape from Earth completely? (190) 9. What is the change in the gravitational potential energy of a 4.00 kg body lifted from Earth’s surface to an altitude of 1.60 × 103 km? Given ME = 6.00 × 1024 kg, RE = 6.40 × 106 m, and G = 6.67 × 10−11 N.m2/kg2. Section 13.4 The Motion of Satellites (191)10. A 1.0 × 103 kg satellite is orbiting Earth at a certain altitude. It takes the satellite 150 minutes to complete one revolution around the Earth. Given ME = 6.00 × 1024 kg, RE = 6.40 × 106 m, and G = 6.67 × 10−11 N.m2/kg2. a. At what altitude is the satellite orbiting above Earth’s surface? b. What is the speed and the angular speed of the satellite? c. Calculate the kinetic energy of this satellite. d. What is the total mechanical energy of the satellite? (192)11. [G] Determine the period of revolution and speed of a satellite moving in a circular orbit of radius 8.0 × 106 m. The mass of Earth is 6.0 × 1024 kg. Section 13.5 Kepler’s Laws and the Motion of Planets (193)12. Mars takes 687 Earth days to orbit the sun. What is the semi-major axis of its orbit? Given MS = 2.00 × 1030 kg and G = 6.67 × 10−11 N.m2/kg2. (194) 13. [G] State Kepler’s three laws of planetary motion. Section 13.6 Spherical Mass Distributions (195)14. A small uniform sphere of mass m is placed at a distance x from one end of a thin uniform rod, of length L and mass M, as shown below. a. Express the gravitational potential energy of the rod-sphere system. Take the gravitational potential energy to be zero when the rod and the sphere are infinitely far apart. Level N | 53  Physics N b. What is the expression of the gravitational potential energy when x is much x1 x 2 x 3 x 4 x 5 larger than L. Use ln (1 + x ) = − + − + −... 1 2 3 4 5 dU c. Using Fx = − , find the magnitude and direction of the gravitational force dx exerted by the rod on the sphere. (196) 15. [T] A student is given the set of orbital data (shown below) for four of the moons of Jupiter and is asked to use the data to determine the mass M of Jupiter. Assume the orbits of these moons are circular. Use G = 6.67 × 10−11 Nm2/kg2. Orbital Period T Orbital Radius R (days) (× 103 km) 7.1546 1,070 16.689 1,883 1.7691 422 3.5512 671 a. Assuming the orbits of Jupiter's moons are circular, derive an equation for the orbital period T of a moon as a function of its orbital radius R. b. Which quantities should be graphed to yield a straight line whose slope could be used to determine Jupiter's mass? c. Complete the data table by calculating the two quantities to be graphed. Convert the values into SI units. Label the top of each column, including units. Level N | 54  Physics N d. Plot the graph on the axes below and draw the best straight-line fit to the points. Label the axes with the variables used and appropriate numbers to indicate the scale. e. Using the graph, calculate the mass of Jupiter. f. Due to a collision with an asteroid, the speed of one of the moons slightly increases. Complete the diagram below to show the new orbit of the moon. (197)16. [T] A geostationary satellite of mass m is launched in a circular orbit whose period of revolution is equal to Earth's period of rotation. a. The radius of the geostationary orbit r can be expressed as r = kRE. Express your answers in terms of the mass of the Earth ME, its radius RE, k, m, and fundamental constants. i. Derive an expression for the speed v of the satellite in a circular orbit. ii. Derive an expression for the speed u with which the satellite is launched into the orbit to acquire a geostationary orbit. ii. Derive an expression for the work done on the satellite by gravity as it moves from the launch site to its orbit. Level N | 55  Physics N b. Calculate the value of the constant k, given the mass of Earth ME = 6.0 × 1024 kg, its radius RE = 6.0 × 106 m, and the period of Earth's rotation TE = 24 h. Use G = 6.67 × 10−11 Nm2/kg2. c. The diagram below shows the predicted shape of the satellite's S orbit around the Earth E. It is observed that the speed of the satellite as it reaches the orbit is slightly smaller than v because of the space debris it encountered on its way up. Complete the diagram to show the path of the satellite on its orbit. Level N | 56  Physics N Chapter 14 Periodic Motion Section 14.1 Describing Oscillation (198) 1. A microwave radiation has a frequency of 500 MHz. a. Calculate the angular frequency of this radiation. b. What is the period of a microwave radiation? (199) 2. [G] An ultrasonic transducer used for scientific investigations oscillates at 10 MHz. How long does each oscillation take, and what is the angular frequency? Section 14.2 Simple Harmonic Motion (200) 3. A certain blade moves back and forth over a distance of 2.0 mm in simple harmonic motion. The frequency of oscillations is 120 Hz. a. What is the amplitude of oscillation? b. Calculate the maximum blade speed. c. What is the magnitude of the maximum blade acceleration? (201) 4. Two identical springs of spring constant 7500 N/m are attached to a block of mass 0.250 kg. The horizontal surface is considered to be frictionless. a. What is the net force exerted by the springs on the block as it is displaced from its equilibrium position? b. Give the equation of motion of the block. c. Calculate the angular frequency of oscillation and deduce its frequency. d. Calculate the period of the oscillations. (202) 5. [G] Define simple harmonic motion and write the equation that describes it. (203)6. [G] The mass of a load suspended from a spring is 0.20 kg. Determine the period of its free vibrations if the force constant of the spring is 80 N/m. How many oscillations will this spring pendulum complete in 11 s? (204) 7. [G] A ball is attached to an oscillating spring. The figure below shows the graph of the ball’s position x as a function of time t. Level N | 57  Physics N What equation describes the displacement of the ball as a function of time? Section 14.3 Energy in Simple Harmonic Motion (205) 8. Consider a spring-mass oscillator of an oscillation amplitude A. a. If the displacement of the block is half the amplitude of oscillation, what fraction of the total energy will the potential energy and kinetic energy be? b. At what displacement is the energy of the system half kinetic and half potential? (206) 9. [G] A small particle of mass 8.0 kg is attached to an ideal spring oscillating in SHM along a horizontal frictionless surface with an angular frequency of 10 rad/s. The amplitude of the motion is 0.20 m. What is the mechanical energy of the system? (207) 10. [G] Describe how the kinetic, potential, and mechanical energy vary in SHM. Section 14.4 Applications of Simple Harmonic Motion (208)11. A sphere, of mass 95 kg and radius 15 cm, is suspended from a massless wire. A torque of 0.20 N.m is required to rotate the sphere through an angular displacement of 0.85 rad. a. What is the torsion constant of this harmonic oscillator? b. What is the period of oscillations that result when the sphere is released? (209) 12. [G] An angler hangs an 80.0 kg fish from an ideal spring having negligible mass. The fish stretches the spring 0.200 m. a. Find the force constant of the spring. b. The fish is now pulled down 5.00 cm and released. What is the angular frequency of oscillation of the fish? Use g = 10 m/s2. (210)13. [G] A balance wheel with moment of inertia of 6 × 10−8 kg∙m2 oscillates with the angular frequency of 6 rad/s. Find the torsion constant. Section 14.5 The Simple Pendulum (211)14. What should the length of a simple pendulum that oscillates with a period of 1.0 s be? Level N | 58  Physics N (212) 15. A simple pendulum of length L oscillates with a period T. If the length of the pendulum is quadrupled, how does the resulting period T’ compare to T? (213)16. [G] What is the length of a simple pendulum performing harmonic oscillations with a frequency of 0.25 Hz on the surface of the Moon? The acceleration due to the gravity on the Moon’s surface is 1.6 m/s2. Section 14.6 The Physical Pendulum (214) 17. A physical pendulum, consisting of a uniform disk of radius 10.0 cm and mass 500 g, is attached to a uniform rod of length 50.0 cm and mass 270 g. a. Calculate the moment of inertia of the physical pendulum about the pivot point. b. What is the distance between the pivot point and the center of mass of the pendulum? c. Calculate the period of oscillations of the physical pendulum. Section 14.7 Damped Oscillations (215) 18. Consider a block of mass 1.50 kg attached to a spring of spring constant 8.00 dx N/m, as shown below. The block is acted upon by a damping force given by −b , dt where b = 230 g/s. The block is pulled down 12.0 cm and then released. Level N | 59  Physics N a. Calculate the time required for the amplitude of the resulting oscillations to be equal to 1/3 the initial amplitude. b. What is the angular frequency of the oscillator in damped motion? c. How many oscillations are made by the block during the period calculated in part a? (216)19. [T] A pendulum is made of a cardboard plate modeled by a uniform solid disk of mass m and radius R hung by a string of length 2R, as shown below. The moment of 1 inertia of a disk rotating about its center is given by I 0 = mR 2. 2 a. The pendulum is allowed to oscillate freely. Express all algebraic answers in terms of the given quantities and fundamental constants. i. By applying the appropriate equation of motion to the pendulum, write the differential equation for the angle θ the pendulum makes with the vertical. ii. By applying the small-angle approximation to your differential equation, calculate the period of the pendulum's oscillation Td. Level N | 60  Physics N An actual plate attached to the string looks like this: b. Describe an experimental procedure that you could use to determine the position of the center of mass of the plate, including the equipment that you would need. c. The plate is now untied from the string, and another much smaller object of the same mass m is tied to it. How does the new period of the pendulum's oscillation Tp compare to the old value Td? ___ Tp = Td ___ Tp > Td ___ Tp < Td Justify your answer. (217) 20. [T] A cart of mass 5m is attached to a spring of constant k. Initially, the spring is not deformed, and the cart is stationary. A sticky ball of mass m is projected horizontally with speed u towards the cart. The ball is initially h meters above the cart. When the ball hits the cart, it sticks to it. Express all algebraic answers in terms of the given quantities and fundamental constants. a. Determine the speed of the cart, V, just after the ball strikes it. State, explicitly, what assumptions you've made in your calculations. b. Derive, but do not solve, the equation of motion of the cart-ball system. c. Determine the period of the simple harmonic motion that results. d. Write the equation of motion of the cart-ball system. Level N | 61  Physics N e. Another ball, of the same mass but made of rubber, is projected from height h horizontally at speed u towards the toy cart. The ball hits the cart and bounces off it, as shown in the diagram below. Indicate below whether the period of the resulting simple harmonic motion of the cart is greater than, less than, or the same as it was in part c. _____ greater _____ less _____ the same Justify your answer. (218)21. [T] A simple pendulum, made of a bob of mass m = 100 g, is performing small oscillations on a light inextensible string of length l = 1.10 m. The pendulum is hanged from a stationary table, and a motion sensor is used to measure the speed of the bob at different moments of time. The data from the sensor is shown in the graph below. a. Determine the period of oscillations of the pendulum. b. Determine the value of g obtained in the experiment. c. The relative error in measuring the length of the pendulum is 2% and the relative error in time measurements is 0.5%. Explain whether the value of g obtained in the experiment is within the experimental error. d. One student suggests that the disparity between the measured and the known value of g is due to the friction in the pendulum's hinge. Explain why this is incorrect. Level N | 62  Physics N (219)22. [T] A spherical, non-rotating planet has a radius R and a uniform density ρ throughout its volume. A narrow tunnel was drilled through the planet along a diameter AB in which a small ball of mass m could move freely under the influence of gravity. Let r be the distance between the ball and the center of the planet. Express your answers in terms of the given and fundamental constants. a. Derive an expression for the magnitude of the force on the ball at a distance r < R from the center of the planet. b. Derive, but do not solve, the differential equation that describes how the position of the ball in the hole varies with time. c. Use the result you obtained in part b to determine the time tB the ball takes to traverse the planet from A to B. d. A satellite of mass m is set into circular orbit of radius R at point A. It takes the satellite tS seconds to fly from A to B. How do the values tS and tB compare? _____ tS < tB _____ tS = tB _____ tS > tB Explain your answer. (220)23. [T] A uniform rod, of mass m and length L, is hung from one of its ends and allowed to perform small oscillations. The angle the meter stick makes with the vertical at a given moment of time is denoted as θ. The rotational inertia of the uniform rod about its end can be expressed as I = nmL2, where n is an unknown constant. Level N | 63  Physics N a. In terms of the quantities given above, derive, but do not solve, the differential equation that could be used to determine the angular displacement θ of the pendulum as a function of time t. b. Express the period of the rod's oscillation T in terms of the given and the fundamental constants. c. Students want to determine the value of n for the uniform rod. To do so, they suspend several rods of different lengths L and measure the time of ten oscillations, t10, for each rod. Their results are summarized in the table below. L (m) t10 (s) 0.2 7.2 0.4 10.3 0.6 12.6 0.8 14.5 1 16.2 i. State which two quantities would produce a straight line on the graph. Complete the table with these values. ii. On the graph below, plot the data points. Draw a straight line that best represents the data. iii. Using g = 9.8 m/s2, determine the value of n. d. The experiment is conducted again in an elevator accelerating upwards at a m/s2. Describe qualitatively how this will affect the slope of the graph obtained in part c (ii). Level N | 64  Physics N Chapter 15 Mechanical Waves Section 15.1 Types of Mechanical Waves (221) 1. Indicate the type of each mechanical wave shown below. Explain your answer. (a) (b) (222) 2. Which of the following physical quantities is transmitted as a wave propagates through a medium? A. density B. volume C. matter D. energy E. mass (223) 3. [G] Define transverse waves and longitudinal waves. Section 15.2 Periodic Waves (224) 4. What is the speed of a tsunami wave that has a wavelength of 200 km and a period of 800 s? (225) 5. Sound waves travel through air at an approximate speed of 340 m/s. What is the wavelength of a sound wave whose frequency is equal to 520 Hz? (226) 6. [G] A boat is swinging in waves with a frequency of 0.50 Hz. What is the speed of these waves if the distance between adjacent crests is 3.0 m? Section 15.3 Mathematical Description of a Wave (227) 7. A transverse wave propagating along a rope is given by y (x, t) = (0.750 cm) cos  [(0.400 cm−1)x + (250 s−1)t]. a. What is the amplitude of the wave? b. What is the wavelength of the wave? c. What is the frequency of the wave? d. What is the speed of propagation of the wave along the rope? e. In what direction is the wave traveling? Level N | 65  Physics N (228) 8. The diagram below shows a periodic wave in a taut string. Consider three points A, B, and C as indicated on the string. A B C Which of the following is true about the wave’s speed at points A, B, and C? A. vA < vB < vC B. vA > vB > vC C. vA = vC > vB D. vA = vB = vC E. vB > vA = vC (229) 9. [G] The wave function is y = 0.2 cos  ( 5t − 2 x ) . Find the amplitude, frequency, wavelength, and speed of propagation of the wave. Section 15.4 Speed of a Transverse Wave (230) 10. Based on the given of question 1 in section 15.3, what is the tension in the rope of linear density 0.050 kg/m? (231)11. Under a tension F, it takes 2.0 seconds for a pulse to travel the length of a taut wire. What tension, in terms of F, is required for the pulse to take 6.0 seconds instead? (232) 12. Two taut strings S1 and S2, having the same mass and length, are set under the same tension. The cross section of S1 is double that of S2. Which of the following is true about the speeds v1 and v2 of transverse waves along S1 and S2, respectively? A. v1 = v2 B. v2 = 2v1 C. v2 = 2v1 D. v2 = 4v1 E. v1 = 2v2 (233) 13. [G] The tension in a string with a linear mass density of 0.03 kg/m is 300 N. Find the speed of a transverse wave on the string. Level N | 66  Physics N Section 15.5 Energy in Wave Motion (234) 14. Based on the given of question 1 in section 15.3, what is the average power of the wave? (235) 15. Two identical but separate strings, under the same tension, carry transverse wav