The actual test contains 1 multiple choice questions and 2 problems. However, for extra exercise, this practice test includes 4 problems. Questions: N.B. Make sure that you justify your answers explicitly on the space provided on this page in order to qualify for partial credit even when your choice is wrong. Q1: [4] A constant force is applied to a box, contributing to a certain displacement on the floor. If the angle between the force and displacement is 135, the work done by this force is a) energy gained by the box. b) energy lost by the box. c) energy gained by the floor. d) energy lost by the floor. e) Both (b) and (c) above are true. Q2: [4] Two boxes, one heavier than the other, are allowed to slide down the same rough incline. It is observed that both of them have the same constant acceleration a. Which of the two boxes has a larger work done by kinetic friction as they reach the bottom of the ramp? a) The heavy one b) The light one c) The friction does the same work in both cases d) The situation described is not physically possible: the mass is different so a should be different e) It depends on the initial speed Force (N) Q3: [4] The adjacent graph represents the net horizontal force acting on an object as it moves in a straight line along x-axis. What is the net work done by this force in the shown interval? a) b) 2 J c) 1 J d) 2 J e) Insufficient information. 1 5 5 1 1 2 3 4 5 θ x (m) Q4: [4] The adjacent graph represents a vertical map with the paths of two balls rolling from rest down two frictionless slopes with the indicated profile. Knowing that ball A is twice as heavy as B, which ball will move faster at the bottom of its hill, and what will be its speed? a) Ball A will move faster, v A = 19.8 m/s. b) Ball A will move faster, v A = 39 m/s. c) Ball B will move faster, v B = 19.8 m/s. d) Ball B will move faster, but the information is insufficient to find the speed. e) Both will move with the same speed, v A = v B =19.8 m/s y (m) 2 15 1 5 A B x (m) 1 2 3 4 5 Q5: [4] The elastic constant k 1 of spring is twice the constant k 2 of spring. The spring is compressed by x 1, and the spring is stretched by x 2. The compression x 1 is also twice the stretch x 2. Then the ratio PE 1 / PE 2 between the elastic potential energies of the two springs is a) 2 b) -2 c) 4 d) 8 e) None of the above. 1
Q6: [4] Two masses, m 1 = 4 kg and m 2 = 1 kg, are attached to identical springs with elastic constant k = 3 N/m, as in the figure. What is the difference between the elastic potential energies stored in the two systems? a) 24 J b) 98 J c).98 J d) 128 J e) None of the above Hint: The elastic potential energies are given by the spring elongations. How would you calculate this elongation at equilibrium? m 1 k m 2 k Q7: [4] Two objects, one heavier than the other, have the same kinetic energy. Which has the smallest linear momentum? a) The light object b) The heavy object c) Both have the same momentum. d) It depends on the potential energy e) It depends of the speed. Q8: [4] Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 2 seconds. Which box has the smaller momentum after the force acts? a) The heavier one. b) The lighter one. c) Depends on the distance traveled in 2 seconds. d) Both have the same momentum. e) Depends on the local gravitational acceleration. Q9: [4] A ball with mass m =.5 kg moves with a speed v i = 4. m/s perpendicularly onto a wall, collides inelastically with the wall and bounces back along the same line with speed v f = 2. m/s. If the ball was in contact with the wall for 1-2 s, how large was the force exerted by the wall on the ball? a) 1 N b) 3 N p i c) N p f d) 9.8 N e) None of the above. Q1: [4] Scrat, the conspicuous Ice Age squirrel, sits on a sled of mass M = 5. kg, holding tightly on his beloved acorn. Scrat s mass is m 1 = 1. kg and acorn s mass is m 2 = 3 g. The sled is at rest when Scrat throws out the acorn horizontally with a speed v 2 = 24 m/s. As a result, the sled moves with a speed a) v 1 = m/s b) v 1 = 24 m/s c) v 1 =.12 m/s d) v 1 = 12 m/s e) Nonsense! Scrat would never throw away his acorn. 2
Problems: In order to qualify for partial credit you have to provide at least a logical start toward a solution, even if it may be flawed. Do not flood the space with obviously useless information. P1: A small box of mass m is launched at position x = with initial horizontal velocity v = 3.5 m/s. The box moves in a straight line along a frictionless surface. Besides the usual forces, an externally applied force acts on the box given by: F F, F A x L, A, x y where the variable x is the position along the x-axis, while A = 4.9 N and L = 4.5 m are constants. y m x a) [5] The box is represented on the figure at position x = L. Figure out the orientation of force F in that position, and sketch the vector force diagram. b) [6] Write out Newton s 2 nd law along the provided x and y-axes. Express the contribution of force F in terms of A, L and x. x-axis: y-axis c) [9] Calculate explicitly the work done by each of the acting forces as the box moves through the interval x = (, L). d) [5] Assuming that the box hovers on the ground (zero normal), use one of the equations in part (b) above to calculate the mass of the box. e) [5] Use the Work-Energy theorem to calculate the speed of the box at position x = L. 3
P2: A box with mass m =.6 kg is used to compress a spring of constant k = 5 N/m by an initial compression x =.12 m. After the spring is released, the box travels successively through configurations shown on the figure. It moves a distance d 1 =.6 m along a horizontal surface. In point, the box starts to climb up a ramp of angle θ = 37 and length d 2 =.5 m. Eventually, the box stops on the ramp in point, at height h with respect to the flat surface. All surfaces are rough, with coefficient of kinetic friction μ k =.17. a) [5] Calculate the mechanical energy E 1 of the box in point. This is the startup energy of the box. m x d 1 v 2 θ d 2 h b) [6] Use the adjacent figure to sketch the vector force diagram for the object in an arbitrary point between. Then calculate the work W fr1 done by the friction between. This is the nonconservative work done on the box in this interval. v c) [6] Use the conservation of energy to calculate the mechanical energy of the box in point. Then use the result to compute the speed v 2 of the box in point. d) [7] Use the adjacent figure to sketch the vector force diagram for the object in an arbitrary point between. Then calculate the work W fr2 done by the friction between. This is the nonconservative work done on the box in this interval. θ e) [6] Use energy conservation to compute the maximum altitude h reached by the box up the ramp (point ). 4
P3: A wooden block of mass M = 1.98 kg is initially at rest when it is hit horizontally by a bullet of mass m = 2. g with speed v = 3 m/s. The bullet is imbedded in the wood and subsequently the bullet-block system compresses a spring with elastic constant k = 2 N/m along a frictionless surface. m v M k a) [5] Denote p bullet, p wood the respective momenta before the collision, and p bullet, p wood after the collision. Write out the momentum conservation during the collision, and use it to calculate the speed of the system bullet-box immediately after the collision. x b) [5] What is the mechanical energy of the system before and after the collision? c) [5] Calculate the percent of energy lost during the collision. What happened with the missing energy? d) [5] Use conservation of mechanical energy to calculate the maximum compression x of the spring. e) [1] Knowing that the maximum compression is achieved in a time Δt =.2 s, what is the average power of the spring? Also, use the Impulse-Momentum Theorem to calculate the average force exerted on the system block-bullet by the spring in this interval. 5
P4: A child plays with a game of identical small disks sliding on a flat frictionless surface. He pushes disk #1 with a velocity v v, v v, (where v = 1.2 m/s) x y 1 1 1 such that it collides with a disk #2 moving with velocity v v, v 1 v, v x y 2 2 2 2 where the velocity components are written in the coordinate system shown on the figure. After the collision, disk #1 moves with velocity v v, v 2 v, 1 v. 1 1x 1y 3 6 Let s calculate the velocity v 2 v 2x, v 2 y of disk #2 after the collision. a) [3] Sketch the net momentum on the figure before and after collision. Then figure out how the momentum ve3ctor p 2 is supposed to look like and draw it on the figure. b) [3] Use the notation for the particle momenta before and after the collision given on the figure and write out the conservation of momentum in this collision. Then write the momenta explicitly in terms of mass and velocities and find the relation between the velocities before and after the collision. y before after x c) [6] Write the velocity relationship you found in part (b) in terms of the components along x and y-directions provided on the figure. Then replace the components of v 1, v 2, and v 1 as given above in terms of constant v, to find two equations. Solve them to find v 2 v 2x, v 2 y. d) [6] Knowing that the mass of each disk is m = 1 g, calculate the net kinetic energy of the disks before and after the collision. e) [2] So, is the collision elastic or inelastic? Explain briefly. 6