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 101. The exam period was 90 minutes; the average score was 73.3; the median score was 77. Click here to see the formula sheet that came with the exam.

A wheel of mass M and radius R = 0.1700 m rests on a horizontal surface against a vertical step of height h = 0.0100 m. A point below the center of the wheel is a distance x = 0.0575 m from the vertical step. The wheel is to be raised over the step by a horizontal force F applied to the axle of the wheel as shown in the figure. Let F_{min} be the minimum force needed to raise the wheel over the step. The figure is not to scale.

What is the magnitude of the torque that the force F_{min} exerts about the point of contact between the wheel and the step?

(a) (0.0200 meters) F_{min} (b) (0.0575 meters) F_{min} (c) (0.1600 meters) F_{min} (d) (0.3200 meters) F_{min} (e) (0.3765 meters) F_{min}

(a) (0.0575 meters) Mg (b) (0.1149 meters) Mg (c) (0.2251 meters) Mg (d) (0.3200 meters) Mg (e) (0.3765 meters) Mg

(a) 0.3379 × Mg (b) 0.3594 × Mg (c) 0.5882 × Mg

(a) T = 42.5 N (b) T = 56.6 N (c) T = 73.5 N (d) T = 127 N (e) T = 147 N

(a) I_{CM} = 2.86 kg-m^{2} (b) I_{CM} = 1.43 kg-m^{2} (c) I_{CM} = 0.71 kg-m^{2}

A beam of mass M and length L is held in the horizontal by two ideal massless ropes that are attached to the ends of the beam. At some time the rope on the left end of the beam breaks.

Before the rope on the left end of the beam breaks, i.e. while the beam is still supported by the two ropes, what is the tension in the rope on the left?

(a) (1/12) Mg (b) (1/4) Mg (c) (1/3) Mg (d) (1/2) Mg (e) (3/4) Mg

(a) α = g / 12L (b) α = g / 4L (c) α = 3g / 2L (d) α = g / 2L (e) α = 3g / 4L

Two blocks are suspended over a pulley with a string of negligible mass as shown below. One block has mass m_{1} = 1 kg, the other one has a mass of m_{2} = 2 kg. The pulley is a uniform solid cylinder. The string does not slip on the pulley and the center of mass of the pulley remains stationary. The downward acceleration a of mass m_{2} is measured to be a = g / 4 .

What is the mass M of the pulley?

(a) M = 1.0 kg (b) M = 1.5 kg (c) M = 1.7 kg (d) M = 2.0 kg (e) M = 2.3 kg

(a) smaller than g/4. (b) equal to g/4. (c) larger than g/4.

A spool of mass M = 1 kg sits on a frictionless horizontal surface. A thread wound around the spool is pulled with a force T = 4 N as shown below. The total moment of inertia about the center of mass of the spool is I = 0.8 kg·m^{2}, its outer radius is R = 1 m and its inner radius is r = 0.5 m. The spool starts from rest.

The magnitude of the angular acceleration α of the spool is:

(a) α = T (r+R) / (I+Mr^{2}) (b) α = T R / I (c) α = T r / I

(T) True (F) False

A solid sphere of mass M and radius R is released from rest on an inclined plane with an angle of θ. The coefficient of static friction for the sphere on the plane is μ_{s}. Assuming that the sphere rolls without slipping down the plane and that the static frictional force is at its maximum value, which of the following is the correct equation for the acceleration of the center of mass of the sphere?

(a) a = g sinθ (b) a = μ_{s}g cosθ (c) a = g sinθ - μ_{s}g cosθ

(a) tanθ (b) (2/5) sinθ (c) (1/5) cosθ (d) (2/7) tanθ (e) (Mg/R) sinθ

A thin hoop of mass M and radius R rolls without slipping up a plane inclined at an angle θ with the horizontal. The initial velocity of the cylinder is V.

The frictional force exerted by the plane on the cylinder is

(a) pointing along the same direction as V. (b) pointing opposite to the direction of V. (c) zero.

(a) H = V^{2} / g (b) H = 2V^{2} / 3g (c) H = 3V^{2} / 4g

(a) H' > H (b) H' = H (c) H' < H

A projectile of mass m = 0.2 kg is flying at the constant velocity V = 10 m/s in the x direction and at a distance of d = 0.4 m from the x-axis towards a solid disk of mass M = 1 kg that is rotating counterclockwise about the fixed axis with the angular velocity ω = 10 rad/s. The projectile strikes the disk and sticks at point A. The radius of the disk is 0.5 m.

What is the angular velocity of the disk, ω_{1}, after the projectile hits the disk?

(a) ω_{1} = 12.3 rad/s (b) ω_{1} = 9.0 rad/s (c) ω_{1} = 6.8 rad/s (d) ω_{1} = 10.0 rad/s (e) ω_{1} = 11.7 rad/s

(a) increasing the final angular velocity ω_{1}. (b) decreasing the final angular velocity ω_{1}. (c) no effect on final angular velocity.

The following questions are based on experimental results obtained from P211 Lab #6. A student measures the position versus time of an object of mass m = 0.132 kg rolling without slipping down an incline. From the measured data she produces the graphs shown below.

Based upon the information provided, what shape did she assume for the rolling object?

(a) solid cylinder (b) hollow cylinder (c) solid sphere (d) hollow sphere (e) There is not enough information to answer this question.

(a) 3.2 m/s (b) 2.1 m/s (c) 1.6 m/s

Two wheels A and B are connected by an ideal massless non-stretchable and non-slipping belt. The radius R_{A} is three times of radius R_{B}.

What would be the ratio of moments of inertia I_{A} / I_{B} if both wheels have the same angular momentum?

(a) I_{A} / I_{B} = 1/3 (b) I_{A} / I_{B} = 1 (c) I_{A} / I_{B} = 3

(a) I_{A} / I_{B} = 9 (b) I_{A} / I_{B} = 1 (c) I_{A} / I_{B} = 3

A woman whose mass is 70 kg stands at the rim of a horizontal turntable that has a moment of inertia 300 kg m^{2} and radius of 2 m. The system is initially at rest and is free to rotate about frictionless vertical axle through the center of the turntable. The woman begins to walk clockwise around the rim at a speed 1.5 m/s relative to the earth.

In what direction and with what angular speed does the turntable rotate?

(a) 0.36 rad/s counterclockwise (b) 0.42 rad/s clockwise (c) 0.70 rad/s counterclockwise (d) 0.42 rad/s counterclockwise (e) 0.70 rad/s clockwise

(a) 44.1 J (b) 78.75 J (c) 152.25 J