General Physics I (Mechanics, Thermal Physics and Waves) PDF

GENERAL PHYSICS I (MECHANICS, THERMAL PHYSICS AND WAVES) Course content; Space and time; units and dimension, kinematics; Fundamental laws of mechanics, statics and dynamics; Work and energy. Conservation laws. Moments and energy of rotation; simple harmonic motion; motion of simple syste...

GENERAL PHYSICS I (MECHANICS, THERMAL PHYSICS AND WAVES) Course content; Space and time; units and dimension, kinematics; Fundamental laws of mechanics, statics and dynamics; Work and energy. Conservation laws. Moments and energy of rotation; simple harmonic motion; motion of simple systems. Temperature; heat, gas laws; Laws of thermodynamics; kinetic Elasticity; Hooke's law, Young's shear and bulk moduli, hydrostatics; Pressure, buoyancy, Archimedes' principles. Surface tension; adhesion, cohesion, capillarity, drops and bubbles. Sound; Types and properties of waves as applied to sound and light energies. Superposition of waves. Propagation of sound in gases, solids and liquids and their properties. The unified spectra analysis of waves. Applications. MECHANICS MECHANICS is the study of the effects of external forces on bodies at rest or in motion. An important objective of Physics is the exact measurement of physical quantities, expressed in units. The unit is a value, quantity, or magnitude in which other values, quantities or magnitudes are expressed. Quantities like length, mass, and time are referred to as fundamental quantities, i.e. base quantities that cannot be expressed in terms of any other physical quantity, having fundamental units of meter (m), kilogram (kg), and second (s) respectively. ASSIGNMENT: (1) List four additional base quantities with their unit and symbols. Answer: Electric Current (Ampere, A), Temperature (Kelvin, K), Amount of Substance (Mole, Mol), and Luminous Intensity (Candela, Cd) (2) Apart from the MKS system (i.e. meter, kilogram, and second), list at least three other systems of units that have been used. Answer: The following systems of units have been in use – (i) The French or C.G.S (Centimeter, Gramme, Second) System; (ii) The British or F.P.S (Foot, Pound, Secons) System; 1 (iii) The M.K.S (Metre, Kilogram, Second) System; and (iv) The S.I. (International System of Units). Dimension Dimension of a physical quantity simply indicates the physical quantities which appear in that quantity and gives absolutely no idea about the magnitude of the quantity. All mechanical quantities, e.g. force, and velocity, can be expressed in terms of three fundamental dimensions namely; length L, mass M, and time T. Therefore, the formula which indicates the relation between the fundamental units and the derived unit is called the dimensional formula. For example, the dimension of acceleration [distance /(time)2] is L/T2 or LT-2, and that of force is MLT-2. Purpose of dimension; (a) To verify if a physical equation is correct Dimension is useful in showing that the following relation for the time period of a body executing simple harmonic motion is correct. where and are the displacement and acceleration due to gravity respectively. Solution; For the relation to be correct, the dimension of L.H.S (left hand side of the equation) must be equal to dimension of R.H.S. (right hand side of the equation). Thus, the dimension of the LHS is M0L0T1 or T. (𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑙)1/2 (𝐿)1/2 (𝐿)1/2 While the dimension of the RHS = = = = (𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑔)1/2 (𝐿𝑇 −2 )1/2 (𝐿)1/2 𝑇 −1 1 =𝑇 𝑇 −1 Note; constant (𝜋) has no dimension, and the values was not also considered. (b) To derive the unit of a physical quantity Example: The viscous drag F between two layers of liquid with surface area of 𝑑𝑣 𝑑𝑣 contact A in a region of velocity gradient is given by 𝐹 = 𝜂𝐴 𝑑𝑥 𝑑𝑥 2 where 𝜂 is the coefficient of viscosity of the liquid. Obtain the unit for 𝜂. 𝐹𝑑𝑥 [𝑀𝐿𝑇 −2 ][𝐿] 𝑀𝐿2 𝑇 −2 Solution: Making 𝜂 the subject of the formula, 𝜂 = = [𝐿2 ][𝐿𝑇 −1 ] = = 𝐴𝑑𝑣 𝐿3 𝑇 −1 𝑀𝐿−1 𝑇 −1 = 𝑘𝑔𝑚−1 𝑠 −1 = unit of 𝜂 Note; distance, 𝑑𝑥 has a dimension of L, and velocity, 𝑑𝑣 has a dimension of 𝐿𝑇 −2 Limitations of dimensional analysis; (i) The method does not provide any information about the magnitude of dimensionless variables and dimensionless constants. (ii) The method cannot be used if the quantities depend upon more than three- dimensional quantities: M, L, and T. (iii) The method is not applicable if the relationship involves trigonometric, exponential and logarithmic functions. Kinematics The study of the motion of objects without referring to what causes the motion, Kinematics. Motion is a change in position in a time interval. The position of an object in space is its location relative to some reference point, often the origin (or zero point). For example, a particle might be located at, which means the position of the particle is in positive direction from the origin. 3 Meanwhile, the displacement is the change from one position to another position. It is given as Speed (scalar quantity), the time interval t to travel to a distance l. 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑒𝑑 𝑙 Average speed = = = 𝑣𝑎𝑣 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑡 The distance, the quantity a car’s odometer reads, is the total length traveled along the path. 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑠 Velocity (vector quantity), 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = = 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑡 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑣𝑓 −𝑣𝑖 Acceleration, 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = = 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑡 Uniform accelerated motion in a straight line (motion in one-dimension): The motion can be described with the five equations for uniformly accelerated motion which includes; 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 (𝑠) = 𝑣𝑎𝑣 𝑡 𝑣𝑓 +𝑣𝑖 𝑣𝑎𝑣 = 2 𝑣𝑓 − 𝑣𝑖 𝑎= 𝑡 𝑣𝑓2 = 𝑣𝑖2 + 2𝑎𝑠 1 2 𝑠 = 𝑣𝑖 𝑡 + 𝑎𝑡 2 Some uniformly accelerated motion examples, if gravitational force and friction is considered zero, include; a bicycle whose brakes have been engaged, a ball rolling down a slope, a ball dropped from the top of a ladder, and a skydiver jumping out of a plane. Although these examples do not maintain absolute uniformity of acceleration, due to interference of gravity and/or friction. Example; A truck’s speed increases uniform from 15 km/h to 60 km/h in 20 s. What is (a) the average speed, (b) the acceleration, (c) the distance traveled, all in units of meters and seconds. 4 The instantaneous velocity describes how fast a particle is moving at a given instant. It is expressed as: Speed is the magnitude of instantaneous velocity; that is, speed is velocity that has no indication of direction either in words or via an algebraic sign. For example, a velocity of +5 m/s or -5 m/s is associated with a speed of 5 m/s. Example: The position of a particle moving on an axis is given by 𝑥 = 7.8 + 9.2𝑡 − 2.1𝑡 3 , with x in meters and t in seconds. What is its velocity at t = 3.5 s? Is the velocity constant, or is it continuously changing? Solution Instantaneous acceleration (or simply acceleration); is the derivative of velocity with respect to time: i.e., the acceleration of a particle at any instant is the second derivative of its position x(t) with respect to time. Example; A particle’s position on the x-axis is given by 𝑥 = 4 − 27𝑡 + 𝑡 3 , with x in meters and 5 t in seconds. (a) find the particle’s velocity function v(t) and acceleration a(t). (b) Is there ever a time when v = 0? Solution FUNDAMENTAL LAWS OF MECHANICS (Newton’s law of motion) Newton’s law of motion includes: 1. Every body continues in its state of rest or uniform motion in a straight line, unless impressed forces act on it. Force is a changer of motion. This law expresses the idea of inertia, reluctance of a body to move and reluctance stop once the body is moving. In the first case, the body is said to be static. While the body is said to be dynamic in the second case. 2. The change of momentum per unit time is proportional to the impressed force, and takes place in the direction of the straight line along which the force acts. If the resultant (or net) force F acting on an object of mass m is not zero, the object accelerates in the direction of the force. Mathematically, Newton’s second law is stated as 𝑑𝑝 𝑑𝑚𝑣 𝑚𝑑𝑣 𝐹= = = = 𝑚𝑎 𝑑𝑡 𝑑𝑡 𝑑𝑡 where F is the total force, p is the momentum, t is the time passed, m is the mass of the object, and a is the acceleration of the object. A skydiver jumps from a plane and accelerates until he reaches the highest velocity possible, when this happens his acceleration is equal to nothing, this happens when air resistance is equal to the downward force of the skydiver. 3. Action and reaction are equal and opposite. In other words, "All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction." When we push a wall, it creates an opposite amount of force, thus, either we move or does the wall move. 6 Ex.; A constant force on a 5.0 kg and reduces its velocity from 7 m/s to 3 m/s in a time of 3 s. Determine the force. 𝑣𝑓 −𝑣𝑖 3−7 −4 𝑎= = = = −1.33 𝑚/𝑠 2 𝑡 3 3 𝐹 = 𝑚𝑎 = (5)(−1.33) = −6.7𝑁 Note: the negative sign indicates that the force is a retarding force, directed opposite to the motion Ex; A force of 10 kgf pulls a sledge of mass 40 kg and overcomes a constant frictional force of 3 kgf. What is the acceleration of the sledge (1kgf = 10 N approx.) Resultant force = 10 kgf – 3 kgf =7kgf = 70 N 𝐹 70 F = ma, 𝑎 = = = 1.75 𝑚/𝑠 2 𝑚 40 Statics and Dynamics Statics; is concerned with the forces that act on bodies at rest under equilibrium conditions. This is expressed in the first part of Newton's first law of motion (A body will remain at rest, zero displacement), where equilibrium conditions are met. Acceleration is always zero in statics, so the right-hand side of the equation of Newton's second law of motion (Fnet = ma) will always amount to zero as well. Dynamics; in mechanics, it studies the forces that cause or modify the movement of an object. It deals with the analysis of physical bodies in motion. Therefore, acceleration is a factor in these problems. Dynamics can be subdivided into Kinematics and Kinetics. WORK AND ENERGY Work is the product of force and distance in the direction of force, 𝑊 = 𝐹𝑑𝑐𝑜𝑠𝜃 𝜃 is the angle between the force and displacement. If F and d are in the same direction, 𝜃 = 0 and W= Fd but it will be -Fd if they are in opposite direction. Force like friction slow down the motion of an object and are then opposite in direction to the displacement. Work is the transfer of energy from one entity to another by way of action of a force applied over a distance. The point of application of the force must move if work is to be done. Work is a scalar quantity, and its unit is joule, J. 7 When an engine does work quickly, it is said to be operating at a high power and conversely, an engine is said to be operating on a low power if it does work slowly. Thus, power is work done per unit second. The unit is joule per second or watt (W) The kilowatt-hour is a unit of energy. If a force is doing work at a rate of 1 kilowatt (1000 J/s), then in 1 hour it will do 1 kW.h of work. 1 kW.h = 3.6× 106 J = 3.6 MJ Energy is given to an object when a force does the work on the object, i.e. a measure of the change imparted to a system. An object that is capable of doing work possesses energy. Energy is a scalar quantity, and its unit is joule, J. When an object possesses an energy because it is moving, the energy is said to be kinetic. The kinetic energy originally possessed by an object that was brought to a rest in a distance s by a constant force F acting againt it is equal to the workdone against F, K.E. = 𝐹 × 𝑠 F = ma, 𝐹 × 𝑠 = mas Recall, 𝑣 2 = 𝑢2 − 2𝑎𝑠, v = 0, a is negative 𝑢2 0 = 𝑢2 − 2𝑎𝑠, = 𝑎𝑠 2 1 Hence, 𝐾. 𝐸. = 𝑚𝑎𝑠 = 𝑚𝑢2 2 Potential Energy: A coiled spring has energy, which is released gradually as the spring uncoils. The energy of the spring is called potential energy, because it arises from the position or arrangement of the body and not from its motion. The energy is equal to the work done in displacing the molecules from their normal equilibrium positions against the forces of attraction of the surrounding molecules. The energy possessed by an object because of the gravitational interaction is called gravitational potential energy. Generally, potential energy = mgh Examples 8 Solution 9 Conservation Laws When the work done in moving round a closed path in a field to the original point is zero, the forces in the field are called conservative forces. If the work done in taking an object round a closed path to its original position is not zero, the force in the field are said to be non-conservative. An example is the case of a wooden block that is pushed round a closed path on a rough floor to its initial position. Thus, work is done against friction, as it moves from the starting position and as it returns. This is contrary to a conservative field in which work is done during part of the path and regained for the remaining part. Another example is when a body falls in the earth’s gravitational field, a small part of the energy is used up in overcoming the resistance of the air. The energy is dissipated/lost as heat (not regained in moving the body back to its original position). This shows that energy may be transformed from one form to another, from mechanical energy to heat, total energy in a given system is always constant (Principle of conservation of energy, and is one of the key principles in science) Recall, force acting on a body is the rate of change of momentum (mv), Change of momentum = mvf - mvi = Ft = impulse of the force on the object So 𝑚𝑣𝑓 − 𝑚𝑣𝑖 𝐹= 𝑡 Considering the linear momentum of objects which collide with each other 𝑚1 𝑢1 + 𝑚2 𝑢2 = 𝑚1 𝑣1 + 𝑚2 𝑣2 The principle of the conservation of linear momentum states that, if no external forces act on a system of colliding objects, the total momentum of the objects remain constant. Example; An object of mass 2 kg is moving with a velocity of 3 m/s and collides heads on with another object of mass 1 kg moving in opposite direction with a with a velocity of 4 m/s. After collision both objects coalesce, so that they move with a common velocity. What is the velocity? Total momentum before collision in the direction of the first object = (2 × 3) − (1 × 4) = 2𝑘𝑔𝑚𝑠 −1 Nb: The momentum of the other object is of opposite sign to the first object. 10 The momentum after collision of the two objects in the direction of the first object = 3v + 1v = 3v Therefore, 3v = 2 2 and 𝑣 = 𝑚𝑠 −1. 3 Moments (of inertia) and Energy of Rotation From Newton’s first law of motion, it is clear that a body has a tendency to remain at rest or in uniform motion. This property of the body is known as inertia. Thus, inertia is that property of a body due to which it opposes or resists any change in its state of rest or uniform motion. The term inertia may be referred to as “the amount of resistance of an object to a change in velocity” or “resistance to change in motion.” This includes changes in the speed of the object or the direction of motion. One aspect of this property is the tendency of things to continue to move in a straight line at a constant speed, when no forces are affecting them. The moment of inertia is defined as the quantity reflected by the body resisting angular acceleration, which is the sum of the product of each particle’s mass and its square of the distance from the axis of rotation. In simpler terms, it is a number that determines the amount of torque required for a certain angular acceleration in a rotating axis. The angular mass or rotational inertia are other names for the moment of inertia. For a rigid object rotating about a fixed point, with a particle of the object making 𝑑𝜃 an angle 𝜃 with a fixed line. The angular velocity, or 𝜔 (radian/seconds), of 𝑑𝑡 every particle about the fixed point is the same for each particle, the velocity v1 = r 𝜔. 1 1 The kinetic energy of A = 𝑚𝑣 2 = 𝑚𝑟 2 𝜔2. 2 2 1 1 For the whole body, the kinetic energy = 𝑚1 𝑟 21 𝜔2 + 𝑚2 𝑟 2 2 𝜔2 +… 2 2 1 1 = 𝜔2 ∑ 𝑚𝑟 2 = 𝜔2 𝐼 2 2 I = moment of inertial of the object about the axis concerned (kgm2). 11  𝑀𝑙 2 For a uniform rod, moment of inertia = , 12 M = mass of the rod l = length of the rod Circular disc, moment of inertia 2𝜋𝑥𝛿𝑥 Moment of Inertia of the ring about an axis through O = ( 𝑀) × 𝑥 2 𝜋𝑎2 12  𝑀𝑎2 And the moment of inertia of the whole disc = 2 1 Moment of inertia of a solid cylinder = 𝑀𝑎2 2 2 Moment of inertia of a sphere = 𝑀𝑟 2 5 The moment of inertia of an object about an axis, ∑ 𝑚𝑟 2 , is sometimes written as 𝑀𝑘 2 , where M is the mass of the object and k is a quantity called the radius of gyration about the axis. The kinetic energy of rotation of a mass whose moment of inertia about an axis 1 is I, and which is rotating about an axis with angular velocity 𝜔, is 𝐼𝜔2 2 The work done on a rotating body during an angular displacement 𝜃 by constant torque 𝜏 is given by 𝑊 = 𝜏𝜃 Analogy of linear and angular quantities; If, in the equations for linear motion, we replace linear quantities by the corresponding angular quantities, we get the corresponding equations for angular motion. Therefore, we have 13 Examples; 14 Simple Harmonic Motion When the bob of a pendulum moves to-and-fro (or reciprocating motion) through a small angle, the bob is said to be moving with simple harmonic motion. Similarly, the prongs of a sounding tuning fork, and the layers of air near it, are moving with simple harmonic motion. So, by definition, simple harmonic motion is the motion of a particle whose acceleration is always (i) directed towards a fixed point, (ii) directly proportional to its distance from that point Figure: Simple harmonic curve The time taken for the foot of the perpendicular to move from C to F and back C, or the time taken by an object to make a full circle is the period (T) 2𝜋 𝑇= 𝜔 where 2𝜋 (𝑟𝑎𝑑𝑖𝑎𝑛𝑠) = 360° is the angle and 𝜔 is the angular velocity. The distance y = ZM = 𝑟𝑠𝑖𝑛𝜃 = 𝑟𝑠𝑖𝑛𝜔𝑡 where 𝜃 = 𝜔𝑡. From the sine graph, the number complete set of values of y from O to Q as a cycle. The number of cycles per second is the frequency (hertz, Hz or cycle/sec). 𝟏 𝒇= 𝑻 Velocity during SHM: The velocity of the object moving round the circle is at A at some instant is 𝑟𝜔, and it is directed along the tangent at A. Thus, the velocity parallel to the diameter FC at this instant is 𝑟𝜔𝑐𝑜𝑠𝜃, by resolving. Recall, 𝑦 = 𝑟𝑠𝑖𝑛𝜃 𝑦2 1 By trigonometric identity, 𝑐𝑜𝑠𝜃 = √1 − 𝑠𝑖𝑛2 𝜃 = √1 − = √𝑟 2 − 𝑦 2 𝑟2 𝑟 15 Therefore, 𝑣 = 𝜔√𝑟 2 − 𝑦 2 = velocity of an object moving with simple harmonic motion. While the maximum velocity, 𝑣𝑚 = 𝜔𝑟, corresponds to y = 0. The acceleration, a, of an object is 𝜔2 𝑦, where y is the distance or displacement of the object from a fixed point, the motion is simple harmonic motion. Maximum acceleration in SHM occurs at the end of the oscillation because the acceleration is directly proportional to the displacement. Hence maximum acceleration is 𝜔2 𝑎, where a is the amplitude and 𝜔 is 2𝜋𝑓𝑜. Cases of SHM; 1. Simple pendulum At B, the force pulling the towards O is directed along the tangent at B, and is equal to 𝑚𝑔𝑠𝑖𝑛𝜃. The tension T, in the wire has no component in this direction. Recall 𝐹𝑜𝑟𝑐𝑒 = 𝑚𝑎𝑠𝑠 × 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛, −𝑚𝑔𝑠𝑖𝑛𝜃 = 𝑚𝑎 The negative sign indicate the force is towards O. when 𝜃 is small, 𝑠𝑖𝑛𝜃 = 𝜃 in radians and 𝜃is also 𝑦/𝑙 Thus 𝑦 −𝑚𝑔𝜃 = −𝑚𝑔 = 𝑚𝑎 𝑙 𝑔 Therefore, 𝑎 = − 𝑦 = −𝜔2 𝑦 𝑙 16  2𝜋 Since , 𝑇 = 𝜔 𝑙 It then implies that 𝑇 = 2𝜋√ 𝑔 2. Spiral spring Hookean law; The increase in length is proportional to the weight provided elastic limit of the spring is not exceeded. A Hookean system (a spring, wire, rod, etc.) is one that returns to itsoriginal configuration after being distorted and then released. Moreover, when such system is stretched a distance x (for compression, x is negative), the restoring force exerted by the spring is given by Hooke’s law, F= -kx The minus sign indicates that the restoring force is always oppoeite in direction to the direction of the displacement. The spring constant (N/m) is a measure of the stiffness of the spring. But Hookes law in terms of the external force F ext = kx The tension (force), T, in a spring is proportional to the extension x produced, T = kx. k is the spring constant. When a mass m is placed on a spring (or an elastic thread) PA of length l suspended from a fixed point P, the spring stretches to O by a lengthe given by T = mg = ke. If the mass is pulled down a little and then released, it vibrates up-and-down above and below O. 𝑘 𝑎=− 𝑥 = −𝜔2 𝑥 𝑚 The minus sign indicates that the diection of a (and F) is always opposite to the direction of the displacement x. Neither a nor F is constant. 2𝜋 𝑚 The motion about O is SHM, and the period, 𝑇 = = 2𝜋√ 𝜔 𝑘 Recall mg = ke 𝑒 Thus 𝑇 = 2𝜋√ 𝑔 17 In practice when the load m is varied and thecorresponding period, T, is measured, a straight line graph is obtained when T2 is plotted against m, thus verifying indirectly that the motion of the load was simple harmonic. The gragh does not pass through the origin, however, owing to the mass and the movement of various parts of the spring. Examples 18 Conversely, when a 400 g mass is hung at the end of a vertical spring, the spring stretches 25 cm. What is the stiffness, and how compressed is the spring if 50 g mass is removed from it. 𝐹𝑒𝑥𝑡 = 𝑘𝑥 𝐹𝑒𝑥𝑡 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑛 𝑡ℎ𝑒 𝑠𝑝𝑟𝑖𝑛𝑔 = 𝑚𝑔 = 0.4 × 9.8 = 3.92 𝑁 𝐹𝑒𝑥𝑡 3.92 So 𝑘 = = = 15.68 𝑁 𝑥 0.25 When 50 g mass is removed, the 𝐹𝑒𝑥𝑡 = (0.4 𝑔 − 0.05 𝑔) × 9.8 = 3.43 𝑁 Thus, using 𝐹𝑒𝑥𝑡 = 𝑘𝑥 𝐹𝑒𝑥𝑡 3.43 the compression (not extension), 𝑥 = = = 0.22 𝑚 𝑘 15.68 A 50 g mass hangs at the of a Hookean spring. When 20 g more is added to the end of the spring, it stretches 7 cm more. (a) Find the spring constant. (b) If 20 g is now removed, what will be the period of the motion? (a)When the 50 g mass hangs at the end of the spring 𝐹𝑒𝑥𝑡1 = 𝑘𝑥1 (1) When 20 g more is added 𝐹𝑒𝑥𝑡1 + 𝐹𝑒𝑥𝑡2 = 𝑘(𝑥1 + 𝑥2 ) (2) Note; 𝐹𝑒𝑥𝑡1 is the weight of 50 g, and 𝑥1 is the stretching it caused while 𝐹𝑒𝑥𝑡2 is weight of 20 g, and 𝑥2 is the stretching it caused. 19 Therefore, when equation 1 is substracted from equation 2, we have 𝐹𝑒𝑥𝑡2 = 𝑘𝑥2 and 𝐹𝑒𝑥𝑡2 0.02 × 9.8 0.196 𝑘= = = = 2.8 𝑁/𝑚 𝑥2 0.07 0.07 𝑚 (b) 𝑇 = 2𝜋√ 𝑘 m is the 50 g So 0.05 𝑇 = 2𝜋√ = 0.84 𝑠 2.8 20 Motion of Simple Systems. A machine is any device through which the magnitude, direction, or method of application of a force is changed so as to achieve some advantage. Typical of such includes lever, inclined plane, pulley, axle, crank, and jackscrew. Based on the principle of conservation, work input in a machine = useful work output + work to overcome friction. In the case of a machine that operate for a short time, some of the work input may be used to store energy within the machine. An internal spring might be stretched, or a movable pulley might be raised, for example. The actual mechanical advantage (AMA) of a machine = force ratio 𝑓𝑜𝑟𝑐𝑒 𝑒𝑥𝑒𝑟𝑡𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑜𝑛 𝑙𝑜𝑎𝑑 Force ratio = 𝑓𝑜𝑟𝑐𝑒 𝑢𝑠𝑒𝑑 𝑡𝑜 𝑜𝑝𝑒𝑟𝑎𝑡𝑒 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 The ideal mechanical advantage (IMA) of a machine = distance ratio 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑚𝑜𝑣𝑒𝑑 𝑏𝑦 𝑖𝑛𝑝𝑢𝑡 𝑓𝑜𝑟𝑐𝑒 Distance ratio = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑚𝑜𝑣𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑙𝑜𝑎𝑑 Due to the presence of friction, the AMA is always less than IMA. Both IMA and AMA are greater than one. 𝑤𝑜𝑟𝑘 𝑜𝑢𝑡𝑝𝑢𝑡 𝑝𝑜𝑤𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 𝐴𝑀𝐴 The efficiency of the machine = = = 𝑤𝑜𝑟𝑘 𝑖𝑛𝑝𝑢𝑡 𝑝𝑜𝑤𝑒𝑟 𝑖𝑛𝑝𝑢𝑡 𝐼𝑀𝐴 Example 21 22