# 1. An object is dropped from rest. Which of the five following graphs correctly represents its motion? The positive direction is taken to be downward.

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1 Unless otherwise instructed, use g = 9.8 m/s 2 Rotational Inertia about an axis through com: Hoop about axis(radius=r, mass=m) : MR 2 Hoop about diameter (radius=r, mass=m): 1/2MR 2 Disk/solid cyllinder about central axis(radius=r, mass=m, length=l): 1/2MR 2 Disk/solid cyllinder about axis perpendicular to length, that is, central diameter (radius=r, mass= M, length=l): 1/4MR 2 +1/12ML 2 Thin rod about axis perpendicular to length (mass=m, length = L): 1/12ML 2 Solid sphere about diameter (radius=r, mass=m): 2/5MR 2 Thin spherical shell about diameter (radius=r, mass=m): 2/3MR 2 Rectangular slab about perpendicular axis (mass=m, sides: a, b): 1/12M(a 2 +b 2 ) Solve the following problems. Enter the answers in your scantron (always use a No 2 pencil. on the scnatron). 1. An object is dropped from rest. Which of the five following graphs correctly represents its motion? The positive direction is taken to be downward. A) I B) II C) III D) IV E) V Page 1

2 2. Two forces, one with a magnitude of 3 N and the other with a magnitude of 5 N, are applied to an object. For which orientations of the forces shown in the diagrams is the magnitude of the acceleration of the object the least? A) I B) II C) III D) IV E) V 3. Three books (X, Y, and Z) rest on a table. The weight of each book is indicated. The force of book Z on book Y is: A) 0 B) 5 N C) 9 N D) 14 N E) 19 N 4. A boat is able to move through still water at 20 m/s. It makes a round trip to a town 3.0 km upstream. If the river flows at 5 m/s, the time required for this round trip is: A) 120 s B) 150 s C) 200 s D) 300 s E) 320 s Page 2

3 5. You stand on a spring scale on the floor of an elevator. Of the following, the scale shows the highest reading when the elevator: A) moves downward with increasing speed B) moves downward with decreasing speed C) remains stationary D) moves upward with decreasing speed E) moves upward at constant speed 6. The velocity of a particle moving along the x axis changes from v i to v f. For which values of v i and v f is the total work done on the particle positive? A) v i = 5m/s, v f = 2m/s B) v i = 5m/s, v f = 2m/s C) v i = 5m/s, v f = 2m/s D) v i = 5m/s, v f = 2m/s E) v i = 2m/s, v f = 5m/s 7. The diagram shows a plot of the potential energy as a function of x for a particle moving along the x axis. The points of stable equilibrium are: A) only a B) only b C) only c D) only d E) b and d Page 3

4 8. A 2-kg block is attached to a horizonal ideal spring with a spring constant of 200 N/m. When the spring has its equilibrium length the block is given a speed of 50 m/s. What is the maximum elongation of the spring? A) 0 B) 0.05 m C) 5 m D) 10 m E) 100 m 9. A particle is released from rest at the point x = a and moves along the x axis subject to the potential energy function U(x) shown. The particle: A) moves to a point to the left of x = e, stops and remains at rest B) moves to the point x = e, then moves to the left C) moves to infinity at varying speed D) moves to x = b where it remains at rest E) moves to x = e and then to x = d, where it remains at rest 10. The center of mass of a system of particles remains at the same place if: A) it is initially at rest and the external forces sum to zero B) it is initially at rest and the internal forces sum to zero C) the sum of the external forces is less than the maximum force of static friction D) no friction acts internally E) none of the above 11. A projectile in flight explodes into several fragments. The total momentum of the fragments immediately after this explosion: A) is the same as the momentum of the projectile immediately before the explosion B) has been changed into kinetic energy of the fragments C) is less than the momentum of the projectile immediately before the explosion D) is more than the momentum of the projectile immediately before the explosion E) has been changed into radiant energy Page 4

5 12. At the same instant that a 0.50-kg ball is dropped from 25 m above Earth, a second ball, with a mass of 0.25 kg, is thrown straight upward from Earth's surface with an initial speed of 15 m/s. They move along nearby lines and pass without colliding. At the end of 2.0 s the acceleration of the center of mass of the two-ball system is: A) 0.25g B) 0.50g C) 0.75g D) g E) g/ A rifle of mass M is initially at rest but free to recoil. It fires a bullet of mass m and velocity v (relative to the ground). After firing, the velocity of the rifle (relative to the ground) is: A) mv B) Mv/m C) mv/m D) v E) mv/m 14. A particle moves along the x axis. Its momentum is graphed below as a function of time. Rank the numbered regions according to the magnitude of the force acting on the particle, least to greatest. A) 1, 2, 3, 4 B) 2, 3, 4, 1 C) 1, 4, 3, 2 D) 1, 3, 4, 2 E) 2, 4, 3, 1 Page 5

6 15. If a wheel turns with constant angular speed then: A) each point on its rim moves with constant velocity B) each point on its rim moves with constant acceleration C) the wheel turns through equal angles in equal times D) the angle through which the wheel turns in each second increases as time goes on E) the angle through which the wheel turns in each second decreases as time goes on 16. The fan shown has been turned off and is slowing as it rotates clockwise. It slows down fast, so that the absolute value of the rate of change of angular velocity dw/dt is comparable to the square of the angular velocity w 2, where w = angular velocity. The direction of the acceleration of the point X on the fan tip could be: A) B) C) D) E) 17. A uniform solid cylinder made of lead has the same mass and the same length as a uniform solid cylinder made of wood (wood, as well as pretty much everything else, has lower density than lead). The rotational inertia of the lead cylinder compared to the wooden one is: A) greater B) less C) same D) unknown unless the radii are given E) unknown unless both the masses and the radii are given Page 6

7 18. A light string of length 3R is attached to the surface of a solid uniform sphere of radius R and mass M and used to suspend the sphere from the ceiling. The rotational inertia of this system about the point of attachment at the ceiling is: A) (2/5)MR 2 B) 9MR 2 C) 16MR 2 D) 47/5MR 2 E) (82/5)MR A meter stick on a frictionless table top is pivoted at the 80-cm mark. The figure below shows the meter stickas viewed from above. A force in the plane of the table is applied perpendicularly to the end of the stick at 0 cm, as shown. A second force (not shown), also on the plane of the table, is applied at the 100-cm end of the stick. If the stick does not rotate: A) for all orientations of B) for all orientations of C) for all orientations of D) for some orientations of for others E) for some orientations of for others Page 7

8 20. A small disk of radius R 1 is mounted coaxially with a larger disk of radius R 2. The disks are securely fastened to each other and the combination is free to rotate on a fixed axle that is perpendicular to a horizontal frictionless table top,as shown in the overhead veiw below. The rotational inertia of the combination is I. A string is wrapped around the larger disk and attached to a block of mass m, on the table. Another string is wrapped around the smaller disk and is pulled with a force as shown. The acceleration of the block is: A) R 1 F/mR 2 B) R 1 R 2 F/(I mr 2 2) C) R 1 R 2 F/(I + mr 2 2) D) R 1 R 2 F/(I mr 1 R 2 ) E) R 1 R 2 F/(I + mr 1 R 2 ) 21. A disk starts from rest and rotates around a fixed axis, subject to a constant net torque. The work done by the torque during the second 5 s is as the work done during the first 5 s. A) the same B) twice as much C) half as much D) three times as much E) four times as much 22. A thin-walled hollow sphere rolls without sliding along the floor. The ratio of its translational kinetic energy to its rotational kinetic energy (about an axis through its center of mass) is: A) 3/2 B) 2 C) 1 D) 1/2 E) 2/3 Page 8

9 23. A forward force acting on the axle accelerates a rolling wheel on a horizontal surface. If the wheel does not slide the frictional force of the surface on the wheel is: A) zero B) in the forward direction and does zero work on the wheel C) in the forward direction and does positive work on the wheel D) in the backward direction and does zero work on the wheel E) in the backward direction and does positive work on the wheel 24. Two disks are mounted on low-friction bearings on a common shaft. The first disc has rotational inertia I and is spinning with angular velocity ω. The second disc has rotational inertia 2I and is spinning in the same direction as the first disc with angular velocity 2ω as shown. The two disks are slowly forced toward each other along the shaft until they couple and have a final common angular velocity of: A) 5ω /3 B) C) D) ω E) 3ω 25. A picture is to be hung from the ceiling by means of two wires. Order the following arrangements of the wires according to the tension in wire B, from least to greatest. A) I, II, III B) III, II, I C) I and II tie, then III D) II, I, III E) all tie Page 9

10 26. A 240-N weight is hung from two ropes as shown. The tension in the horizontal rope has magnitude: A) 0 B) 656 N C) 480 N D) 416 N E) 176 N 27. A particle is in simple harmonic motion with period T. At time t = 0 it is at the equilibrium point. At which of the following times is it furthest from the equilibrium point? A) 0.5T B) 0.7T C) T D) 1.4T E) 1.5T 28. The displacement of an object oscillating on a spring is given by x(t) = x m cos(ω t + φ ). If the initial displacement is zero and the initial velocity is in the negative x direction, then the phase constant φ is: A) 0 B) π/2 radians C) π radians D) 3π/2 radians E) 2π radians 29. A 3-kg block, attached to a spring, executes simple harmonic motion according to x = 2cos(50t) where x is in meters and t is in seconds. The spring constant of the spring is: A) 1 N/m B) 100 N/m C) 150 N/m D) 7500 N/m E) none of these Page 10

11 30. Suppose we have three different uniform spheres, all pivoted on horizontal axes that are tangent to their surfaces. Sphere X has radius R and mass M. Sphere Y has radius R and mass 2M. Sphere Z has radius 2R and mass M. Rank the spheres according to their periods of oscillation, least to greatest. A) X and Z tie, then Y B) Y, Z, X C) Z, X, Y D) X, then Y and Z tie E) X and Y tie, then Z Page 11

12 Answer Key - final-seca 1. B 2. A 3. C 4. E 5. B 6. E 7. B 8. C 9. B 10. A 11. A 12. D 13. C 14. B 15. C 16. D 17. B 18. E 19. A 20. C 21. D 22. A 23. D 24. A 25. D 26. D 27. B 28. B 29. D 30. E Page 12

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