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 110. The exam period was 90 minutes; the average score was 71.1; the median score was 74. Click here to see the formula sheet that came with the exam.

A mass M hangs from the end of a rod of mass m and length L. The rod is held in place with a hinge attached to a wall on the right, and it has a cable with a tension which is measured to be T.

What is the value of M, in terms of m, L,T and θ ?

(a) M = (T / (gsinθ)) - m/2 (b) M = (T sinθ/g) - m/2 (c) M = m tanθ (T / g - m) / (T/g + m) (d) M = T cosθ + mg (e) M = m sinθ + T / g

(a) decreases. (b) stays the same. (c) increases.

A solid cylinder (I = 1/2 MR^{2}) and a solid sphere (I = 2/5 MR^{2}), of the same radius and mass, roll down an incline of angle θ = 30°. They both start from rest at a distance D = 2 meters up the incline, as shown in the diagram, and roll without slipping to the bottom of the incline.

The initial gravitational potential energy of the two objects is partially lost in work done overcoming frictional forces as the objects roll down the incline.

(T) True (F) False

(a) larger for the solid sphere than it is for the solid cylinder. (b) the same for both objects. (c) smaller for the solid sphere than it is for the solid cylinder.

(a) 0.76 (b) 0.83 (c) 0.91 (d) 1.04 (e) 1.17

Four 1 kg-masses are held together in the form of a square by four rigid, massless rods, each of length 2l. The positions of the point masses are designated a, b, c and d, as shown in the diagram. Axis AB lies in the plane formed by the square and passes through its center. Axis CD is parallel to AB and passes through the masses c and d. Axis XY is perpendicular to the plane of the square and passes through its center, as shown on the right.

Compare the three moments of inertia of the square about the three specified axes, I_{AB}, I_{CD}, I_{XY}. Which one of the following statements is correct?

(a) I_{AB} = I_{CD} > I_{XY} (b) I_{AB} < I_{CD} < I_{XY} (c) I_{AB} < I_{CD} = I_{XY } (d) I_{AB} > I_{CD} < I_{XY} (e) I_{AB} < I_{CD} > I_{XY}

(a) 14.7 kg l ^{2} (b) 18.6 kg l ^{2} (c) 19.2 kg l ^{2} (d) 21.7 kg l ^{2} (e) 24.9 kg l ^{2}

A massless rope is holding a solid disk of mass M and radius R (I = 1/2 MR^{2}). The rope is attached to the center of the disk and a frictionless, vertical wall as shown in the figure. The center of the disk is at a distance L below where the rope is attached to the wall.

What is the force exerted on the disk by the wall?

(a) Mg (b) MgL (c) MgR/L (d) sqrt(2)MgL (e) MgR

(a) T_{disk} > T_{sphere} (b) T_{disk} = T_{sphere} (c) T_{disk} < T_{sphere}

Consider the earth as a uniform, solid sphere of radius R_{e} = 6.4 × 10^{6} m with mass M_{e} = 6.4 × 10^{24} kg that makes one complete revolution in 1 day (= 24 hours.) As part of a carefully crafted attack, aliens from outer space drop a small neutron star on the equator. The mass of the neutron star is M_{n} = 6.4 × 10^{24} kg, the same as the mass of the earth. (Assume the neutron star is a point mass.)

After the mass is dropped, how long does it take the earth to make one complete revolution?

(a) 0.93 days (b) 1.00 days (c) 2.00 days (d) 2.67 days (e) 3.50 days

(a) It would be less than with the star on the equator. (b) It would be the same as with the star on the equator. (c) It would be greater than with the star on the equator.

A spool with a thread wound around it is pulled with a force T = 30 N as shown below. The total moment of inertia of the spool is I = 1.25 kg·m^{2}, its mass is M = 10 kg, its outer radius is R = 0.5 m and its inner radius is r = 0.1 m. The spool rolls without slipping and starts from rest.

Find the angular acceleration α of the spool.

(a) α = 1.60 rad/s^{2} (b) α = 2.28 rad/s^{2} (c) α = 2.95 rad/s^{2} (d) α = 3.32 rad/s^{2} (e) α = 4.80 rad/s^{2}

After the rockets shut off, what will be the rotational frequency of the saucer?

(a) ω = 0.078 rad/s (b) ω = 0.90 rad/s (c) ω = 1.59 rad/s (d) ω = 2.20 rad/s (e) ω = 2.74 rad/s

(a) L = 7 m (b) L = 12 m (c) L = 15 m (d) L = 18 m (e) L = 20 m

A piece of gum of mass m = 0.1 kg, is thrown at a bar of mass M = 1 kg, and length d = 1 m, pivoted about its center and initially at rest as shown. It sticks to the end of the bar and the final angular speed of the bar is measured to be ω = 3 rad/s. Assume that the gum has a size much smaller than d.

What is the initial speed v of the gum?

(a) v = 3.7 m/s (b) v = 4.8 m/s (c) v = 5.2 m/s (d) v = 5.9 m/s (e) v = 6.5 m/s

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

Of the following choices, which best describes the approximate location of the center of mass of the bar?

(a) At the junction between the dark-shaded and light-shaded pieces. (b) At the pivot. (c) In the light-shaded region, to the right of the pivot.

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

Calculate the distance x if the tension T_{1} in the first rope is measured to be 85 N.

(a) x = 0.44 m (b) x = 0.36 m (c) x = 0.80 m (d) x = 0.60 m (e) x = 0.25 m

A turntable has a mass of 1 kg and a radius of 0.17 m and is initially rotating freely at 78 rpm (ω_{i,t} = 8.168 rad/s). There are no external torques acting on the system. The moment of inertia of the turntable can be approximated by that of a disk (I_{disk} = MR^{2}/2).

A small object, initially at rest, is dropped vertically onto the turntable and sticks to the turntable at a distance d of 0.10 m from its center as shown in the figure. When the small object is rotating with the turntable, the angular velocity of the turntable ω_{f,t} is 72.7 rpm (7.613 rad/s). What is the mass of the small object that was dropped onto the turntable?

(a) 0.048 kg (b) 0.070 kg (c) 0.086 kg (d) 0.105 kg (e) 0.123 kg

(a) 0.080 J (b) 0.095 J (c) 0.103 J (d) 0.121 J (e) 0.137 J

(a) less than that of placing the first record on the turntable. (b) zero. (c) more than that of placing the first record on the turntable.