Concept #1: Torque & Acceleration (Rotational Dynamics)

Practice: Suppose that piano has a long, thin bar ran through it (totally random), shown below as the vertical red line, so that it is free to rotate about a vertical axis through the bar. You push the piano with a horizontal 100 N (blue arrow), causing it to spin about its vertical axis with 0.3 rad/s^{2} . Your force acts at a distance of 1.1 m from the bar, and is perpendicular to a line connecting it to the bar (green dotted line). What is the piano’s moment of inertia about its vertical axis?

Concept #2: Torque & Acceleration of a Point Mass

A mass, m1 = 680 g hangs from on end of a string that goes over a pulley with a moment of inertia of 0.0125 kg.m2 and radius of 15.0 cm. A mass, m 2 = 355 g hangs from the other end. When the masses are released, the larger mass accelerates downward, the lighter mass accelerates upward, and the pulley turns without the string slipping on the pulley. What is the acceleration of the masses? (Hint: this will require three applications of Newton’s 2nd Law-each mass and the pulley!)

A uniform rod of mass m and length l is pivoted about a horizontal, frictionless pin at the end of a thin extension (of negligible mass) a distance l from the center of mass of the rod. The rod is released from rest at an angle of θ with the horizontal, as shown in the figure.
What is the magnitude of the Horizontal force Fx exerted on the pivot end of the rod extension at the instant the rod is in a horizontal position? The acceleration due to gravity is g and the moment of inertia of the rod about its center of mass is 1/12 mℓ2.
1. Fx = 1/13 mg sin(θ)
2. Fx = 24/13 mg cos(θ)
3. Fx = 24/13 mg sin(θ)
4. Fx = 13/12 mg cos(θ)
5. Fx = 12/13 mg cos(θ)
6. Fx = 12/13 mg sin(θ)
7. Fx = 13/12 mg sin(θ)
8. Fx = mg cos(θ)
9. Fx = 1/13 mg cos(θ)
10. Fx = mg sin(θ)

A 3.0kg bucket is attached to a disk-shaped pulley (I = MR 2/2) of radius 12.0 cm and mass 60kg. Suppose the bucket is allowed to fall.
a) Find the linear acceleration a of the bucket.
b) Find the angular acceleration ω of the pulley.
c) How far does the bucket drop in 3.0 s (starting from rest)?
d) What is the tension T in the rope?

A wheel of radius R = 0.20 m is mounted with frictionless bearings about an axle through its center. A light rope is wrapped around the wheel and an object is suspended from the free end of the rope. When the system is released from rest, the object descends with linear acceleration a = 3.0 m/s2 and the tension in the rope is 50 N. What is the angular acceleration of the wheel?

A wheel of radius R = 0.20 m is mounted with frictionless bearings about an axle through its center. A light rope is wrapped around the wheel and an object is suspended from the free end of the rope. When the system is released from rest, the object descends with linear acceleration a = 3.0 m/s2 and the tension in the rope is 50 N. What is the moment of inertia of the wheel for rotation about the axis through its center?

A large door 2.0 m wide is initially at rest. The door is free to turn about frictionless hinges along one edge. A person applies a constant force F = 120 N perpendicular to the door at the edge opposite from the hinges, so a distance of 2.0 m from the line along the hinges. The force remains perpendicular to the door as the door rotates. The door turns through 90° in 4.0 s. What is the moment of inertia of the door for an axis along the hinges?

A uniform disk with mass 40 kg and radius 0.20 m is provided at its center about a horizontal frictionless axle. The disk is initially at rest and then a constant force of F = 30 N is applied tangent to the rim of the disk. What is the magnitude of the resultant linear acceleration, a, of a point on the rim of the disk after the disk has turned through 0.20 revolutions?

A thin-walled hollow sphere with radius R = 0.050 m is released from rest at the top of an incline, a vertical distance of 2.0 m above the bottom of the incline. The moment of inertia of the sphere about the rotation axis through its center is (2/3) mR2. There is sufficient friction for the sphere to roll without slipping. What is the angular velocity of rotation of the sphere when it gets to the bottom of the incline?

If the torques on an object add up to zero,
A) the object is at rest.
B) the object cannot be rotating, but might have a translational motion.
C) the object could have a translational acceleration but it could not be rotating.
D) the forces on the object also add up to zero.
E) the object could be rotating and have a translational acceleration.

A uniform bar (I = 1/3 ML2 for an axis at one end) has mass M = 8.00 kg and length L = 6.00 m. The lower end of the bar is attached to a wall by a frictionless hinge. The bar is held at an angle of 30.0° above the horizontal by a wire that is attached between the upper end of the bar and the wall. The wire makes an angle of 53.1° with the bar. If the wire breaks, what is the initial acceleration (in rad/s 2) of the bar, just after the wire breaks?

A large wheel with radius R is mounted on a frictionless axle that passes through the center of the wheel. A light rope is wrapped around the wheel and a block is suspended from the free end of the rope. When the system is released from rest, the block has a downward acceleration of magnitude 5.00 m/s2 and the tension in the rope as the block descends is 60.0 N. R = 0.30 m. What is the mass of the block?

A solid uniform sphere with mass 3.00 kg and radius 0.400 m is released from rest at the top of a ramp that is inclined at 36.9° above the horizontal. The sphere rolls without slipping as it moves down the ramp. As it moves down the ramp, what is the acceleration of its center of mass?

A pulley of mass M and radius R pivots freely about its center with Icm = kM R 2, where k is some numerical constant. A string is attached to a mass m and run over the pulley as in the sketch. If a downward force F is applied by your hand to the string, find a, the acceleration of the block in terms of F, m, M, k and g only. Take upward to be positive.
1. a = F + mg/ (k + 1) m
2. a = F - mg/ kM - m
3. a = F - mg/ (k + 1) M
4. a = F - mg/ (kM + m)
5. a = F - mg/ kM - m
6. a = F - mg/ (k + 1) m
7. a = F - mg/ (k - 1) M
8. a = F + mg/ (k + 1) M
9. a = F - mg/ (k - 1) m
10. a = F + mg/ kM + m

A uniform disk of radius 0.3 m is mounted on a frictionless, horizontal axis. A light cord wrapped around the disk supports a 0.64 kg object, as shown. When released from rest the object falls with a downward acceleration of 3.5 m/s2.What is the mass of the disk? The acceleration due to gravity is 9.8 m/s 2.A. 1.33391B. 2.304C. 4.28649D. 3.93333E. 1.9008F. 1.66909G. 3.5237H. 1.56255I. 5.61167J. 3.16522

A wheel of radius R = 0.20 m is mounted with frictionless bearing about an axle through its center. A light rope is wrapped around the wheel and an object of mass m = 5.0 kg is suspended from the free end of the rope. When the system is released from rest, the block descends with linear acceleration a = 2.0 m/s2. While the block is descending, what is the tension in the rope?

If a constant net torque is applied to an object, that object will
A) having a decreasing moment of inertia.
B) rotate with constant linear victory.
C) rotate with constant angular velocity.
D) rotate with constant angular acceleration.
E) having an increasing moment of inertia.