(A) What is the angular velocity (in rad /s) and the linear velocity (in m/s) of the edge of the wheel ?
Ans: {34.3 m/s, 74.5 rad/s}

(B) How long will it take the wheel to rotate through 79.3^o ?

Ans: {18.6 ms}

BONUS:
(C) If a 21.0 cm long arrow is shot at the wheel (parallel to the axis of rotation), what is the minimum speed that the arrow must have to get through the spokes ? Assume that the width of each spoke is negligible. If the widths of the spokes are negligible, does it matter where between the hub and the rim the arrow is aimed ?
Ans: {19.9 m/s, ?}

(A) what is its angular deceleration ?

Ans: {a = -241 rad/s ²}

(B) how long did it take to come to rest ?

Ans: {t = 1.83 sec}

26-1. The initial angular velocity of the flywheel is 380 rpm. The angular acceleration of a flywheel decays exponentially according to

(A) Determine w(t), the angular velocity as a function of time. Hint you will need to integrate a( t) to find w(t).

Ans: {w(t) = 85.0 rad/s - (45.2 rad/s) e^-t/t}

(B) What is the angular velocity of the flywheel after 5.00 s ?

Ans: {55.6 rad/s}

(C) Determine q(t), angular rotation as function of time ? Assume that q(0) = 0.

Ans: {q(t) = (85.0 rad/s) t + (525 rad)(e^-t/t -1)}

(D) What is the average angular velocity during the first 15.0 s ?

Ans: {w_av = 59.6 rad/s}

(A) A 450 gram {300 g} solid yellow-striped ball with a radius of 4.75 cm {3.00 cm} spinning at 15.2 revolutions per second {2.50 rev/s} about its center.
Ans: {I = 1.08x10^-4kg-m², w = 15.7 rad/s, RE = .0133 J}

(B) A 375 kg {400 kg} solid granite disk of radius 12.5 cm revolving about a point 5.80 cm from it center. If it takes the disk 1.45 s {2.50 s} to make one complete revolution.
Ans: {I = 4.47 kg-m², w = 2.51 rad/s, RE = 14.1 J}

(C) A 2.50 m {3.70 m} long thin bar of mass 780 grams rotates about one end with a 420 g bead of inconsequential size attached 1.75 m from the pivot point. When the end of the rod is given an initial speed of 220 cm/s ?

Ans:{I_sys = 4.85 kg-m², w = .595 rad/s, RE = .857 J}

(A) What is the moment of inertia of the bar about point A {B} ?

Ans: {I = .0916 kg-m²}

(B) What is the net torque that is acting on the bar ? Note that the weight of the bar also exerts a torque on the bar when it is not pivoted about its center of mass.

Ans: {t _net = -2.557 + 1.458 + 0 + 1.036 + 0 = -.0628 N-m, clockwise}

(C) What is the angular acceleration of the bar initially ?

Ans: {a = -.686 rad/s², clockwise}

BONUS:

(A) how long and how many revolutions will the grindstone make before the grindstone comes to a halt ?

Ans: {3.63 sec, 2.97 rev}

29. A thin cord is wrapped around the outside of a solid disk with a mass of 11.0 kg and radius of 13.0 cm. The cord is pulled horizontally by a constant force of 4.72 N {42.9 N} applied to the cord. If the disk starts from rest and rolls without slipping it will have and constant angular acceleration of 4.40 rad/s ² {40.0 rad/s ²}.

(A) What is the linear velocity of the center of mass of the disk 3.70 seconds after it starts rolling ?

Ans:{19.2 m/s}

(B) What is the net torque acting about the center of mass of the disk ?
Ans:{3.72 N·m}

(C) What is the magnitude and direction of the force of friction acting on the disk due to its contact with the surface ?

Ans:{14.3 N to the right}

(D) What is the net work done on the disk after 3.70 seconds ?

Ans:{3.05 kJ}

BONUS:

(E) If the coefficient of static friction is equal to .520 {.480}, what minimum applied force will cause the disk to slip rather than roll without slipping ?

Ans:{155 N}

30. A woman with a mass of 47.0 kg {62.0 kg} stands on the rim of a horizontal turntable with a moment of inertia of 530 kg-m² and radius of 3.40 m {2.50 m}. The system is initially at rest, and the turntable is free to rotate about its center on frictionless bearings. The woman takes off from rest and runs around the rim in counterclockwise direction at a constant speed of 1.50 m/s relative to the ground .

(A) After the woman starts running, what is the total angular momentum of the woman and the turntable ?
Ans: {L_total = 0}

(B) What is the angular momentum of the table after the woman starts running ?

Ans: {L_T = -233 kg-m ²/s}

(C) What is the angular velocity of the table and its direction after the woman starts running ?

Ans: {w_T = -.439 rad/s, clockwise}

(D) What is the minimum amount of work that the woman must do to set herself and the turntable in motion in part (C) ?

Ans: {W = 121 J}

(E) How fast is the woman running on the surface of the turntable in m/s (i.e. what is her speed relative to the surface of the turntable ?

(F) If she stops running and slides to a stop on the turntable, how fast will the turntable be moving ?

BONUS:

(G) If the woman runs at 1.50 m/s on the ground, tangent to the rim of the turntable, and then jumps onto the rim of the turntable, what would be the turntable's final angular velocity in this situation ? Assume the turntable is at rest before she jumps on and that she comes to a rest on the turntable after she is on the turntable.

31. A 3.80 m long bar with a mass of 2.60 kg is pivoted about one end by a frictionless barring. Initially the bar is horizontal when it is released, rotating counterclockwise. When the bar reaches the bottom of its arc it collides with a 1.90 kg block resting on a frictionless horizontal tabletop. After the collision the block is observed to be moving to the right at speed of 5.50 m/s {3.50 m/s}. ( This problem involves both the conservation of angular momentum and the conservation of linear momentum.)

(A) How much angular momentum is lost during this collision ?
Ans:{None}

(B) What is the angular momentum of the bar just before it collides with the block ?

Ans:{34.8 kg m ²/s}

(C) What is the magnitude and direction of the angular velocity of the bar just after the collision ?

Ans:{.762 rad/s, CCW}

(D) How much energy is lost during this collision ?
Ans:{33.1 J}

BONUS:

(E) If the collision last for 23.0 ms, what average force does the rod exert on the block ?

Ans:{289 N}