a. Determine the potential energy of the spring as a function of displacement.

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1 PSI AP Physics C Work and Energy (With Calculus) Free Response Problems Use g = 10 m/s 2 1. A spring is found with a force that doesn t obey Hooke s Law: F = -kx 2. This spring is placed on a horizontal frictionless surface with one end fastened to a wall and the other end is connected to a bock of mass m. The spring is stretched a distance A and is then released. Present all results in terms of k, A, and m. a. Determine the potential energy of the spring as a function of displacement. b. Determine the maximum potential energy of the spring. c. Determine the maximum speed of the block. d. Determine the location where the potential energy equals the kinetic energy.

2 2. A spring with a spring force given by the formula: F = -α x 1 2, where x is the distance from equilibrium that the spring is compressed or stretched and α is a constant. A group of physics students used the spring on an experiment to investigate the elastic properties of the spring. They recorded the collected data in the table below. Force (N) Compression (m) a. On the diagram below, make a graph that can be used to determine the constant A. Be sure to label the x and y axes with the appropriate variable. b. Find the constant α from the graph. c. Develop an expression for the potential energy that can be stored in the spring. d. The spring is used in a spring gun placed horizontally at the edge of a 0.75 m tall table. The spring is compressed to a potential energy of 1.8 J. It then launches a 0.5 kg dart. Find the maximum distance from the table where a dart projected from the gun hits the floor.

3 3. An object is acted on by a force with a magnitude Fx = and moves along the x axis. a. Calculate the work done by the external force when the object moves from point x1 to point x2. b. Calculate the work done by the external force when the object moves from point x2 to point x1. c. What is the difference between the answers in part (a) and (b)? What is the net work done on the object by the force as it moves from x1 to x2 and then back to x1? x 4. A spring is placed on a horizontal table with one end fixed. The spring doesn t obey Hooke s Law and in order to stretch or compress it we need to apply the following force: Fx = (150 N/m)x (650 N/m 2 )x 2 + (12,500 N/m 3 )x 3. a. Calculate the work required to stretch the spring from the equilibrium position to x = 0.1 m. b. Calculate the work required to compress the spring from the equilibrium to x = -0.1 m. c. Compare your answers in part (a) and (b). 5. An object with a mass of 5 kg is initially at rest on a frictionless horizontal surface. A horizontal force is applied to the object. The object changes its position as: x = (0.4 m/s 2 )t 2 + (0.04 m/s 3 )t 3. a. Calculate the velocity of the object at t = 5 s. b. Calculate the kinetic energy of the object at t = 5 s. c. Calculate the force at t = 5 s. d. Calculate the work done by the force on the object during the first 5 s.

4 6. A spring exerts a restoring force Fx = -ax bx 2, when it is stretched or compressed, and a = 50 N/m and b = 20 N/m 2. The mass of the spring is negligible. a. Calculate the potential energy of the spring as a function of position assuming that U(x = 0) = 0. b. Calculate the work done by the spring when it stretches from x = 0 to x = 0.5 m. c. An object with a mass of 0.5 kg is attached to the free end of the spring and is placed on a frictionless horizontal table. The object is pulled to a distance 0.5 m and released. Find the speed of the object when it is located at x = 0.3 m from the equilibrium point. 7. A force exerted on an object is Fx = -αx 2, where α = 15 N/m 2. a. How much work is done by the force on the object when it moves along a straight line from x = 0 to x = 0.4 m? b. How much work is done by the force on the object when it moves from point x = 0.4 to point x = 0? c. Is this force conservative? Explain. d. If the force is conservative, find the potential energy, assuming that U(x = 0) = An object moving in the xy plane is acted on by a conservative force. The potential energy of the object is described by the formula: U(x,y) = a(x 2 + y 2 ) + bxy, where a and b are positive constants. a. Find the x component of the applied force. b. Find the y component of the applied force. c. Present the force in terms of unit vectors (i and j ).

5 PSI AP Physics C Work and Energy (With Calculus) Free Response Problems Answer-Key: 1. a. U = 1 3 kx3 b. U max = 1 3 ka3 c. v = 2 3 ka 3 m 2. a. 10 F x 1 2 b. α 25 Nm 2 c. 2α 3 x3 2 d m

6 3. a. 1 x 2 1 x 1 b. 1 x 1 1 x 2 c. (a) is negative, while (b) is positive. The net work done is zero. 4. a. 0.85J b. 1.28J c. a. is 0.43 less than (b). 5. a. 7 m/s b J c. 10 N d J 6. a. 25x x3 b. -7.1J c m/s 7. a J b J c. Yes. Work in both directions is equal in magnitude. d. x a. -2ax by b. -2ay bx c. ( 2ax by)i + ( 2ay bx)j