True-false questions are worth 2 points each, three-choice multiple choice questions are worth 3 points each, five-choice multiple choice questions are worth 6 points each. The maximum possible score is 102. The exam period was 90 minutes; the average score was 76.5; the median score was 81. Click here to see the formula sheet that came with the exam.

A pair of forces, F_{1} and F_{2} of magnitude F is applied tangentially to a spool to create a torque about its symmetry axis as shown below. The spool has ends in the form of disks, each of mass M and radius R. The center of the spool is also a disk, with radius r and of negligible mass.

What is the moment of inertia I_{s} of the spool about its axis of symmetry?

(a) I_{s} = Mr^{2} (b) I_{s} = MR^{2} (c) I_{s} = MR^{2} / 2

(a) α = FR / I_{s} (b) α = Fr / I_{s} (c) α = 2Fr / I_{s}

(a) I_{A} = I_{s} + 9MR^{2} / 2 (b) I_{A} = I_{s} + 3MR^{2} / 4 (c) I_{A} = I_{s} + M(R^{2}/4 + r^{2}/4) (d) I_{A} = I_{s} + M(3R^{2}/2 + r^{2}/4) (e) I_{A} = I_{s} + 3MR^{2} / 2

Consider the following simple object made up of two uniform spheres of radius r and mass m, connected by a massless stick. The three axes are mutually perpendicular. Axis "1" passes along the stick; axes "2" and "3" pass through the center of mass of the system.

What is the relationship among the three moments of inertia of the system, calculated with respect to the three axes, I_{1}, I_{2} and I_{3}?

(a) I_{1} = I_{2} = I_{3} (b) I_{1} < I_{2} < I_{3} (c) I_{1} < I_{2} = I_{3}

(a) t/τ about axis 2 (b) t*τ about axis 2 (c) t*τ about axis 1 (d) t*τ*I_{2} about axis 2 (e) τ*I_{2}/t about axis 2

Two masses, m_{1} = 3 kg and m_{2} = 2 kg, are suspended with a massless rope over a pulley of mass M = 10 kg. The pulley turns without friction and may be modeled as a uniform disk of radius R = 0.1 m. You may neglect the size of the masses. The rope does not slip on the pulley. The system begins at rest.

If m_{1} starts at a height h = 1 m above the ground, what is its speed v when it hits the ground?

(a) v = 3.82 m/s (b) v = 2.97 m/s (c) v = 2.04 m/s (d) v = 1.40 m/s (e) v = 0.981 m/s

(a) larger. (b) the same. (c) smaller.

(a) zero because the tension in each rope is the same so there is no torque. (b) greater than zero because its kinetic energy increases. (c) less than zero because the torque on the pulley and the angular displacement point in opposite directions.

(a) (b) (c)

(a) 1.32 m/s (b) 5.29 m/s (c) 6.91 m/s (d) 8.33 m/s (e) 9.81 m/s

A uniform beam of length L = 3 meters and mass M = 150 kg has a block of mass m = 300 kg and negligible size firmly attached to one end as shown below. The beam is attached by a hinge to the wall and extends perpendicularly from the wall. It is braced with a rope attached some distance away, which makes an angle θ = 30° with respect to the beam.

What is the tension T in the rope?

(a) T = 3679 newtons (b) T = 4185 newtons (c) T = 6590 newtons (d) T = 7358 newtons (e) T = 8309 newtons

(a) along the beam, pointing in towards the hinge (b) up and to the right, perpendicular to the rope but in the plane of the page (c) out of the page.

(a) |α| = 1.00 radians/sec^{2} (b) |α| = 1.83 radians/sec^{2} (c) |α| = 2.10 radians/sec^{2} (d) |α| = 2.71 radians/sec^{2} (e) |α| = 3.50 radians/sec^{2}

A 25 kg boy sits on a see-saw with his 85 kg father. The see-saw consists of a 2 m long board of mass 20 kg and a fulcrum which can be placed anywhere along the board to provide the best balance. Neglect the sizes of the boy and his father.

When the see-saw is balanced, how far away is the boy from the fulcrum?

(a) 0.15 m (b) 0.73 m (c) 1.00 m (d) 1.46 m (e) 1.87 m

(a) 441 N (b) 833 N (c) 1274 N

The gyroscope pictured is turning such that the top of the wheel is coming out of the page, and the bar (which is attached to the center of mass of the gyroscope) is sitting on a fulcrum.

The angular momentum vector of the gyroscope points in which direction?

(a) +x (b) -x (c) +z

(a) counterclockwise in the plane of the paper (rotating in the direction from the x axis to the y axis). (b) out of the paper (rotating in the direction from the x axis to the z axis). (c) into the paper (rotating in the direction from the x axis to the -z axis).

(a) (b) (c) (d) (e)

A bar of mass M = 2 kg and length L = 0.4 m lies at rest on a horizontal frictionless floor. One end of the bar is attached to a fixed frictionless pivot A.

A putty ball of mass m = 0.2 kg moving at a velocity of 2 m/s perpendicular to the bar strikes the other end of the bar and sticks to it as shown in the figure.

What is the angular velocity ω of the bar after the putty ball strikes it?

(a) ω = 1.15 s^{-1} (b) ω = 1.68 s^{-1} (c) ω = 1.98 s^{-1} (d) ω = 2.65 s^{-1} (e) ω = 2.88 s^{-1}

(a) It will be bigger. (b) It will be the same. (c) It will be smaller.

(a) KE_{1} > KE_{2} (b) KE_{1} = KE_{2} (c) KE_{1} < KE_{2}

A fly of mass 2 g (.002 kg - it's a big fly), viewed from above, begins to walk counter-clockwise on a turntable of mass 0.5 kg. The turntable is a uniform cylinder of radius 0.2 m that turns without friction about its center, and the fly is situated half way between the center and the outer radius of the cylinder. The fly and the turntable are initially at rest.

If the fly's speed is .005 m/s in its circular path (relative to the ground), what is the magnitude of the angular velocity of the turntable?

(a) 2.5 × 10^{-6} s^{-1} (b) 7.0 × 10^{-6} s^{-1} (c) 2.5 × 10^{-5} s^{-1} (d) 1.0 × 10^{-4} s^{-1} (e) 3.2 × 10^{-4} s^{-1}

(a) still rotating clockwise. (b) not rotating at all. (c) rotating counter-clockwise.