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 113. The exam period was 90 minutes. The mean score was75.2; the median was 75. Click here to see page1 page2 of the formula sheet that came with the exam.

Unless told otherwise, you should assume that the acceleration of gravity near the surface of the earth is 9.8 m/s^{2} downward and ignore any effects due to air resistance.

A solid spherical ball with mass M = 7 kg, radius R = 0.3 m, and initial velocity v_{i} = 5 m/s rolls without slipping up a ramp and reaches the top. (Recall I = 2/5 M R^{2} for a solid uniform sphere.)

Calculate L_{i}, the initial angular momentum of the ball around its rotation axis before it goes up the ramp.

(a) L_{i} = 3.57 kg m^{2}/s (b) L_{i} = 4.20 kg m^{2}/s (c) L_{i} = 8.62 kg m^{2}/s (d) L_{i} = 13.3 kg m^{2}/s (e) L_{i} = 35 kg m^{2}/s

(a) L_{i} < L_{f} (b) L_{i} = L_{f} (c) L_{i} > L_{f}

(a) E = 67.5 J (b) E = 84.3 J (c) E = 98.1 J (d) E = 123 J (e) E = 152 J

(a) v_{f} = 1.49 m/s (b) v_{f} = 3.15 m/s (c) v_{f} = 3.82 m/s (d) v_{f} = 4.10 m/s (e) v_{f} = 5.00 m/s

You have your bicycle upside down for repair. The front wheel is free to rotate about its axis and is perfectly balanced except for the 0.025 kg valve stem 0.3 m from the rotation axis. The location of the stem is at an angle θ with respect to the horizontal, as shown in the figure. Gravity points downward in the figure.

For which of the following values of θ is the torque about the wheel’s axis due to the weight of the stem the smallest?

(a) 0° (b) 45° (c) 90°

(a) 0.176 Nm (b) 0.363 Nm (c) 0.014 Nm (d) 0.067 Nm (e) 0.214 Nm

A uniform board of length L and weight W is supported in two places: one on the extreme left end and the other 2L/3 from the left end. What is the force exerted by the right support on the board?

(a) 3W/4 (b) W/2 (c) 2W/3 (d) W (e) 3W/2

(a) W/4 (b) W/2 (c) W (d) 2W (e) 4W

Which force results in the largest torque about an axis perpendicular to the plane of plane of the board and passing through its center?

(a) (b) (c)

(a) object A (b) object B (c) Both objects have the same moment of inertia about the x axis.

A star is a uniform sphere of mass M and radius R. It rotates about its center with angular velocity ω. The star then collapses to radius R/10 (with the mass still M). What is the angular velocity of the star after its collapse? Note that there are no external torque on the star during the collapse.

(a) ω / 100 (b) ω / 10 (c) ω (d) 10 ω (e) 100 ω

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

A 0.125 kg ball is dropped from a height h above the ground and is observed to be traveling downward with a speed of 7 m/s just before hitting the ground. It bounces back up to a height of 2 meters. Neglect any energy losses due to the friction with the air.

From what height h was the ball dropped?

(a) h = 1.5 m (b) h = 2.1 m (c) h = 2.5 m

(a) F = 13 N (b) F = 18 N (c) F = 24 N (d) F = 28 N (e) F = 37 N

An 85 kg bobsled starts from rest at the top of a frictionless course with a net 80 meter vertical drop. The bobsledder starts by running and pushing the sled with a constant force F over the first 15 meter flat section. The bobsled is observed to be traveling 45 m/s when it reaches the bottom of the run (before applying the brakes to stop).

What is the minimum average force F the bobsledders must have applied over those 15 meters to accelerate the bobsled at the start of the run?

(a) 1300 N (b) 1800 N (c) 4300 N

(a) 71 m (b) 90 m (c) 116 m (d) 138 m (e) 156 m

Two blocks are sliding to the right on a frictionless surface and collide as shown below. Before the collision, the block M on the left has a speed of 4 m/s, and block 3M has a speed of 1 m/s. After the collision the right block (3M) is observed to travel to the right with speed 3 m/s.

What is the velocity of the 1M-block after the collision?

(a) 2 m/s to the left (b) 5 m/s to the left (c) 0 m/s (d) 2 m/s to the right (e) 5 m/s to the right

(T) True (F) False

A 75 kg student (represented by the solid black circle) standing at R=0.8 m from the center of a merry-go-round that is rotating with angular velocity ω = 2.5 radians/second. The merry-go-round alone has a moment of inertia of 599 kg-m^{2}.

What is the speed of the student as he goes around standing on the rim of the merry-go-round?

(a) 2.0 m/s (b) 3.5 m/s (c) 4.3 m/s

(a) 0.36 (b) 0.51 (c) 0.69 (d) 0.72 (e) 0.93

(a) 0.124 radians/s^{2} (b) 0.750 radians/s^{2} (c) 1.23 radians/s^{2}

An external torque t is applied to the disk and the student standing on it. The torque slows down the rotation a constant angular acceleration of α = -65 rad/s^{2}. What is the magnitude of the applied torque? That is, what is the absolute value of the average torque on the merry-go-round while it is slowing down?

(a) τ = 21 N-m (b) τ = 119 N-m (c) τ = 3710 N-m (d) τ = 40070 N-m (e) τ = 42060 N-m

(a) W_{f} = -34 J (b) W_{f} = -15 J (c) W_{f} = -7.5 J (d) W_{f} = -0.8 J (e) W_{f} = 0 J

Two disks are traveling on a frictionless surface and collide. Before the collision, the 1M disk is traveling 3 m/s in the positive x direction, and the 2M disk is traveling 2 m/s in the positive y direction as shown below. The two disks collide and stick together. (Gravity is perpendicular to the plane of the frictionless surface, i.e. perpendicular to the x-y plane)

Calculate the x component of the velocity of the two disks after the collision.

(a) v_{x} = 1 m/s (b) v_{x} = 2 m/s (c) v_{x} = 3 m/s

(a) v_{y} = 1 m/s (b) v_{y} = 1.3 m/s (c) v_{y} = 2 m/s