Forces Pages 26-35 PDF

# Topic 2: Mechanics ## How Scientists Study Mechanics * **Acceleration and velocity are both vector quantities, having both magnitude(size) and direction.** * **Acceleration of a car can be produced by a change in the magnitude of the car's velocity (speed) or by a change in the direction of...

# Topic 2: Mechanics ## How Scientists Study Mechanics * **Acceleration and velocity are both vector quantities, having both magnitude(size) and direction.** * **Acceleration of a car can be produced by a change in the magnitude of the car's velocity (speed) or by a change in the direction of the car's velocity.** **Example:** A car traveling at 12.0 meters/second east that accelerates at 2.0 meters/second² east for 4.0 seconds, increases its speed by 8.0 meters/second. **Example:** On the other hand, a car traveling at 12.0 meters/second east that accelerates at 2.0 meters/second² west for 4.0 seconds decreases the car's speed by 8.0 meters/second. **Example:** A car traveling along a horizontal circular path experiences acceleration directed toward the center of the circular path. This acceleration is called centripetal acceleration. Although the car may be traveling at a constant speed, the car's direction of travel continuously changes and thus the car is accelerating. ## Vocabulary | Term | Definition | | ----------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | Acceleration | The time rate of change of velocity | | Centripetal acceleration | The acceleration of a body moving in a circular path that is directed toward the center of that path | | Centripetal force | The force that acts perpendicular to the motion of an object, keeping it moving in a circular path | | Closed system | A group of objects not acted on by any external force | | Coefficient of friction | The constant of proportionality between the force of friction and the normal force between two objects in contact. The coefficient of kinetic friction between surfaces in contact while the surfaces are in motion | | Displacement | The vector representing the change in position from an object's initial position to its final position | | Distance | The length of the path an object travels | | Equilibrium | The state of an object that is at rest or moving with a constant velocity | | Free fall | The motion of a body caused by gravitational force only | | Free-body diagram | A diagram that represents all the forces acting on an object at a point in time | | Friction | The force that opposes the relative motion of two surfaces or objects in contact. | | Gravitational field | The region around any object with mass that would exert a force on another mass placed in that region. | ## Kinematics The branch of physics that deals with forces and the way they produce and change motion is called mechanics. Kinematics is the mathematical treatment of the motions of bodies without regard to the forces that produce the motion. ## Distance And Displacement * **Distance:** Total length of the path traveled. * **Displacement:** Change in position from an object's initial position to its final position. * **Distance is a scalar quantity.** * **Displacement is a vector quantity.** **Example:** If a car travels on the NYS Thruway from Buffalo to Albany to New York City. The distance traveled by the car is approximately 418 miles or 673 kilometers. The magnitude of the total displacement of the car, however, is only the length of the vector connecting Buffalo and New York City - approximately 313 miles or 504 kilometers. ## Speed and Velocity * **Speed:** Distance traveled in a given time period. * **Velocity:** Time rate of change of displacement. * **Speed is a scalar quantity.** * **Velocity is a vector quantity.** **Example:** If one car travels at 88 kilometers per hour due east and a second car travels at 88 kilometers per hour due north, both cars have the same speed. However, the velocities of the cars differ because the direction of travel is not the same. ## Linear Motion * **Linear motion refers to an object's change of position along a straight line.** **Graphs of Linear Motion** * **Position vs. Time** * The slope of a position versus time graph at any point equals the object's speed at that instant. * A straight line indicates constant velocity. * A straight horizontal line represents zero velocity - an object at rest. * If a position-time graph is a curved line, the velocity is not constant. * **Velocity vs. Time** * The slope of a velocity versus time graph at any point equals the object's acceleration. * A horizontal line with zero slope indicates constant speed or no acceleration. * A straight line with a positive slope indicates increasing speed or constant acceleration. * A straight line with a negative slope indicates decreasing speed or constant negative acceleration. ## Acceleration * **Acceleration:** The time rate of change of velocity. * **Acceleration is a vector quantity.** * **Formula:** _a = Δv/t_ **Example:** A particle is accelerated uniformly from rest to a speed of 50. meters per second in 5.0 seconds. The average speed of the particle during this 5.0-second time interval is _v = (v₁ + v₂) / 2_ _v = 25 m/s_ ## A Projectile Fired Horizontally * **If air resistance is neglected, the object's horizontal velocity remains constant.** * **The initial vertical velocity of the object is zero but the vertical velocity increases as the object accelerates downward due to gravity.** ## A Projectile Fired At An Angle * **A projectile's motion can be studied by resolving the initial velocity into its horizontal and vertical components.** * **If air resistance is ignored, The horizontal component of the velocity remains constant.** Using the appropriate equations of motion, you can calculate things like maximum height, range, and the time to reach maximum height. ## Uniform Circular Motion The force acting on an object moving in a circular path is called the centripetal force. ## Centripetal Acceleration * **Centripetal acceleration is directed towards the center of the circular path.** * **Formula:** _ac = v²/r_ ## Centripetal Force * **Centripetal force is the force needed to keep an object moving in a circular path.** * **Formula:** _F = mv²/r_ ## Newton's Universal Law of Gravitation Every body in the universe exerts a force of attraction on every other body. * **Formula:** _Fg = Gm₁m₂/r²_ **Example:** Calculate the magnitude of the gravitational force of attraction that Earth exerts on the Moon. _Fg = Gm₁m₂/r²_ _Fg= (6.67 × 10-11 N•m²/kg²)(5.98 × 10²⁴ kg)(7.35 × 10²² kg)/(3.84 × 10⁸ m)²_ _Fg = 1.99 × 10²⁰ N_ ## Gravitational Field Strength * **A region in space where a test particle would experience a gravitational force is called a gravitational field.** * **Formula:** _g = F/m_ ## Weight * **The gravitational force with which a planet attracts a mass is called weight.** * **Formula:** _F = mg_ **Example:** The weight F of a 1.00-kilogram object on Earth's surface is 9.81 newtons. ## Impulse and Change in Momentum * **Impulse:** The product of the net force acting on an object and the time during which the force acts. * **Formula:** _J = Fnett_ * **Impulse is a vector quantity.** * **Impulse can be determined graphically by finding the area under the curve of a force vs. time graph.** **Example:** A 5.0-kilogram object has an initial velocity of 8.0 meters per second due east. An unbalanced force acts on the object for 3.0 seconds, causing its velocity to decrease to 2.0 meters per second east. Calculate the magnitude and direction of the unbalanced force. _F = mΔv/t_ _F = (5.0 kg)(2.0 m/s - 8.0 m/s)/3.0 s_ _F = -10 N_ The force is 10 N directed to the west. ## Conservation of Momentum * **A group of interacting objects that are not acted upon by any external force is called a closed system. ** * **Law of conservation of momentum:** The total momentum of the objects in a closed system is constant. * **Formula:** _Pbefore = Pafter_ ## The Simple Pendulum * **A simple pendulum consists of a bob or mass m attached to a string of negligible mass.** * **The period of a simple pendulum is the time required for one complete cycle of motion** * **Formula:** _T = 2π√(L/g)_ ## Equilibrium and Nonequilibrium Forces When a pendulum is in its equilibrium position, the net force on the bob is zero. If the bob is displaced from equilibrium, the pendulum is no longer in equilibrium. ## Static and Kinetic Friction * **Static friction:** The force that opposes the start of motion. * **Kinetic friction:** The force that opposes the motion between objects in contact. * **Coefficient of static friction:** The coefficient of friction when the objects are at rest. * **Coefficient of kinetic friction:** The coefficient of friction when the objects are in motion. ## Friction on an Inclined Surface If an object is on a surface inclined at angle θ to the horizontal, the object's weight can be resolved into two components, one perpendicular to the inclined surface and the other parallel to the surface. The perpendicular component of the object's weight, F, cos θ or mg cos θ, is equal in magnitude and opposite in direction to the normal force and has no effect on the motion of the object. The object cannot move in the direction of either force. The component of the object's weight parallel to the inclined surface F, sin θ or mg sin θ tends to accelerate the object down the incline. As the angle that the incline makes with the horizontal increases, the component of the object's weight parallel to the incline increases, and the acceleration of the object down the incline increases. This acceleration is opposed by the friction between the object and the incline. The magnitude of the force of friction is directly proportional to the normal force, which is equal in magnitude but opposite in direction to the perpendicular component of the objects's weight. ## Fluid Friction Fluid friction depends upon the surface area and the speed of the object moving through the fluid. ## Momentun The product of an object's mass and velocity is a vector quantity called momentum. * **Formula:** _p = mv_ ## Impulse and Change in Momentum * **Impulse:** The product of the average net force acting on an object and the time during which force acts. * **Formula:** _J = Fnett_ ## The Simple Pendulum A simple pendulum consists of a bob or mass m attached to a string of negligible mass. The length of the pendulum l is measured from the pivot point at one end of the string to the center of the bob, where all the mass is assumed to be concentrated. In the equilibrium position, the string is perpendicular to the ground. To set the pendulum in motion, the bob is displaced from the equilibrium position by lifting it in the gravitational field. The angle the string makes with the equilibrium position is called the amplitude, θ. ## Period of a Simple Pendulum * **Formula:** _T = 2π√(l/g)_ ## Equilibrium and Nonequilibrium Forces When a pendulum is in the equilibrium position, two forces act on the bob, the weight F, and the tension in the string Fr. The tension is equal in magnitude and opposite in direction to the weight, so there is no net force on the bob. If the bob is displaced from equilibrium and the pendulum has an amplitude θ, the pendulum is no longer in equilibrium. The tension in the string is still directed along the string, but it is not opposite in direction to the weight. If the weight of the bob is resolved into perpendicular components Fox and Fgy, the tension in the string is found to be less than the weight of the bob. The net force on the bob, equal to the component of its weight, Fox, acts along the tangent to its path. This net force causes the bob to accelerate towards its equilibrium position. ## Static and Kinetic Friction There are several kinds of friction. Static friction is the force that opposes the start of motion, whereas kinetic friction is the friction between objects in contact when they are in motion. Once motion starts, kinetic friction decreases. The force of kinetic friction for two surfaces in contact is less than the force of static friction for the same two surfaces, so the coefficient of kinetic friction is less than the coefficient of static friction. For example, according to the *Reference Tables for Physical Setting/Physics*, the coefficient of kinetic friction for copper on steel is 0.36 and the coefficient of static friction for copper on steel is 0.53. ## Friction on an Inclined Surface If an object is on a surface inclined at angle θ to the horizontal, the object's weight can be resolved into two components, one perpendicular to the inclined surface and the other parallel to the surface. The perpendicular component of the object's weight, F, cos θ or mg cos θ, is equal in magnitude and opposite in direction to the normal force and has no effect on the motion of the object. The object cannot move in the direction of either force. The component of the object's weight parallel to the inclined surface F, sin θ or mg sin θ tends to accelerate the object down the incline. As the angle that the incline makes with the horizontal increases, the component of the object's weight parallel to the incline increases, and the acceleration of the object down the incline increases. This acceleration is opposed by the friction between the object and the incline. The magnitude of the force of friction is directly proportional to the normal force, which is equal in magnitude but opposite in direction to the perpendicular component of the objects's weight. ## Fluid Friction Fluid friction depends upon the surface area and the speed of the object moving through the fluid. ## Momentun The product of an object's mass and velocity is a vector quantity called momentum. * **Formula:** _p = mv_ ## Impulse and Change in Momentum * **Impulse:** The product of the average net force acting on an object and the time during which the force acts. * **Formula:** _J = Fnett_ ## The Simple Pendulum A simple pendulum consists of a bob or mass m attached to a string of negligible mass. The length of the pendulum l is measured from the pivot point at one end of the string to the center of the bob, where all the mass is assumed to be concentrated. In the equilibrium position, the string is perpendicular to the ground. To set the pendulum in motion, the bob is displaced from the equilibrium position by lifting it in the gravitational field. The angle the string makes with the equilibrium position is called the amplitude, θ. ## Period of a Simple Pendulum * **Formula:** _T = 2π√(l/g)_ ## Equilibrium and Nonequilibrium Forces When a pendulum is in the equilibrium position, two forces act on the bob, the weight F, and the tension in the string Fr. The tension is equal in magnitude and opposite in direction to the weight, so there is no net force on the bob. If the bob is displaced from equilibrium and the pendulum has an amplitude θ, the pendulum is no longer in equilibrium. The tension in the string is still directed along the string, but it is not opposite in direction to the weight. If the weight of the bob is resolved into perpendicular components Fox and Fgy, the tension in the string is found to be less than the weight of the bob. The net force on the bob, equal to the component of its weight, Fox, acts along the tangent to its path. This net force causes the bob to accelerate towards its equilibrium position. ## Static and Kinetic Friction There are several kinds of friction. Static friction is the force that opposes the start of motion, whereas kinetic friction is the friction between objects in contact when they are in motion. Once motion starts, kinetic friction decreases. The force of kinetic friction for two surfaces in contact is less than the force of static friction for the same two surfaces, so the coefficient of kinetic friction is less than the coefficient of static friction. For example, according to the *Reference Tables for Physical Setting/Physics*, the coefficient of kinetic friction for copper on steel is 0.36 and the coefficient of static friction for copper on steel is 0.53. ## Friction on an Inclined Surface If an object is on a surface inclined at angle θ to the horizontal, the object's weight can be resolved into two components, one perpendicular to the inclined surface and the other parallel to the surface. The perpendicular component of the object's weight, F, cos θ or mg cos θ, is equal in magnitude and opposite in direction to the normal force and has no effect on the motion of the object. The object cannot move in the direction of either force. The component of the object's weight parallel to the inclined surface F, sin θ or mg sin θ tends to accelerate the object down the incline. As the angle that the incline makes with the horizontal increases, the component of the object's weight parallel to the incline increases, and the acceleration of the object down the incline increases. This acceleration is opposed by the friction between the object and the incline. The magnitude of the force of friction is directly proportional to the normal force, which is equal in magnitude but opposite in direction to the perpendicular component of the objects's weight. ## Fluid Friction Fluid friction depends upon the surface area and the speed of the object moving through the fluid. ## Momentun The product of an object's mass and velocity is a vector quantity called momentum. * **Formula:** _p = mv_ ## Impulse and Change in Momentum * **Impulse:** The product of the average net force acting on an object and the time during which the force acts. * **Formula:** _J = Fnett_ ## The Simple Pendulum A simple pendulum consists of a bob or mass m attached to a string of negligible mass. The length of the pendulum l is measured from the pivot point at one end of the string to the center of the bob, where all the mass is assumed to be concentrated. In the equilibrium position, the string is perpendicular to the ground. To set the pendulum in motion, the bob is displaced from the equilibrium position by lifting it in the gravitational field. The angle the string makes with the equilibrium position is called the amplitude, θ. ## Period of a Simple Pendulum * **Formula:** _T = 2π√(l/g)_ ## Equilibrium and Nonequilibrium Forces When a pendulum is in the equilibrium position, two forces act on the bob, the weight F, and the tension in the string Fr. The tension is equal in magnitude and opposite in direction to the weight, so there is no net force on the bob. If the bob is displaced from equilibrium and the pendulum has an amplitude θ, the pendulum is no longer in equilibrium. The tension in the string is still directed along the string, but it is not opposite in direction to the weight. If the weight of the bob is resolved into perpendicular components Fox and Fgy, the tension in the string is found to be less than the weight of the bob. The net force on the bob, equal to the component of its weight, Fox, acts along the tangent to its path. This net force causes the bob to accelerate towards its equilibrium position. ## Static and Kinetic Friction There are several kinds of friction. Static friction is the force that opposes the start of motion, whereas kinetic friction is the friction between objects in contact when they are in motion. Once motion starts, kinetic friction decreases. The force of kinetic friction for two surfaces in contact is less than the force of static friction for the same two surfaces, so the coefficient of kinetic friction is less than the coefficient of static friction. For example, according to the *Reference Tables for Physical Setting/Physics*, the coefficient of kinetic friction for copper on steel is 0.36 and the coefficient of static friction for copper on steel is 0.53. ## Friction on an Inclined Surface If an object is on a surface inclined at angle θ to the horizontal, the object's weight can be resolved into two components, one perpendicular to the inclined surface and the other parallel to the surface. The perpendicular component of the object's weight, F, cos θ or mg cos θ, is equal in magnitude and opposite in direction to the normal force and has no effect on the motion of the object. The object cannot move in the direction of either force. The component of the object's weight parallel to the inclined surface F, sin θ or mg sin θ tends to accelerate the object down the incline. As the angle that the incline makes with the horizontal increases, the component of the object's weight parallel to the incline increases, and the acceleration of the object down the incline increases. This acceleration is opposed by the friction between the object and the incline. The magnitude of the force of friction is directly proportional to the normal force, which is equal in magnitude but opposite in direction to the perpendicular component of the objects's weight. ## Fluid Friction Fluid friction depends upon the surface area and the speed of the object moving through the fluid. ## Momentun The product of an object's mass and velocity is a vector quantity called momentum. * **Formula:** _p = mv_ ## Impulse and Change in Momentum * **Impulse:** The product of the average net force acting on an object and the time during which the force acts. * **Formula:** _J = Fnett_