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Physics 1120 In-Class Problems: Rotational Dynamics

    Pulleys

  1. Three point masses lying on a flat frictionless surface are connected by massless rods. Determine the angular acceleration of the body (a) about an axis through point mass A and out of the surface and (b) about an axis through point mass B. Express your answers in terms of F, L, and M. You will need to calculate the moment of inertia in each case.

  2. The object in the diagram below is on a fixed frictionless axle. It has a moment of inertia of I = 50 kg-m2. The forces acting on the object are F1 = 100 N, F2 = 200 N, and F3 = 250 N acting at different radii R1 = 60 cm, R2 = 42 cm, and R3 = 28 cm. Find the angular acceleration of the object.

  3. A rope is wrapped around a solid cylindrical drum. The drum has a fixed frictionless axle. The mass of the drum is 125 kg and it has a radius of R = 50.0 cm. The other end of the rope is tied to a block, M = 10.0 kg. What is the angular acceleration of the drum? What is the linear acceleration of the block? What is the tension in the rope? Assume that the rope does not slip.

  4. Two blocks are connected over a pulley as shown below. The pulley has mass M and radius R. What is the acceleration of the blocks and the tension in the rope on either side of the pulley? (HINT: The tension must be different or the pulley would not rotate.)

  5. A winch has a moment of inertia of I = 10.0 kg-m2. Two masses M1 = 4.00 kg and M2 = 2.00 kg are attached to strings which are wrapped around different parts of the winch which have radii R1 = 40.0 cm and R2 = 25.0 cm.
    (a) How are the accelerations of the two masses and the pulley related?
    (b) Determine the angular acceleration of the masses. Recall that each object needs a separate free body diagram.
    (c) What are the tensions in the strings?

  6. A rope connecting two blocks is strung over two real pulleys as shown in the diagram below. Determine the acceleration of the blocks and angular acceleration of the two pulleys. Block A is has mass of 10.0 kg. Block B has a mass of 6.00 kg. Pulley 1 is a solid disk, has a mass of 0.55 kg, and a radius of 0.12 m. Pulley 2 is a ring, has mass 0.28 kg, and a radius of 0.08 m. The rope does not slip.

    Rolling Objects

  7. A yo-yo has a mass M, a moment of inertia I, and an inner radius r. A string is wrapped around the inner cylinder of the yo-yo. A person ties the string to his finger and releases the yo-yo. As the yo-yo falls, it does not slip on the string (i.e. the yo-yo rolls). Find the acceleration of the yo-yo.

  8. A solid cylinder rolls down an inclined plane without slipping. The incline makes an angle of 25.0 to the horizontal, the coefficient of static friction is µs = 0.40, and Icyl = ½MR2. Hint - you may not assume that static friction is at its maximum!
    (a) Find its acceleration.
    (b) Find the angle at which static friction is at its maximum, at just above this angle the object will start to slip.

  9. A thin-shelled cylinder rolls up an inclined plane without slipping. The incline makes an angle of 25.0 to the horizontal, the coefficient of static friction is µs = 0.40, and Ihoop = MR2.
    (a) Find its acceleration.
    (b) Find the angle which the object will start to slip.

  10. A toy car has a frame of mass M and four wheels of mass m. The wheels are solid disks. The car is placed on an incline and let go. Assume each tire supports one-quarter of the car's weight.
    (a) Find the acceleration of the toy car.
    (b) If the coefficient of static friction is µ, find an expression for the angle at which the wheels begin to slip.

  11. A person pulls a heavy lawn roller by the handle with force F so that it rolls without slipping. The handle is attached to the axle of the solid cylindrical roller. The handle makes an angle θ to the horizontal. The roller has a mass of M and a radius R. The coefficients of friction between the roller and the ground are μs and μk.
    (a) Find the acceleration of the roller.
    (b) Find the frictional force acting on the roller.
    (c) If the person pulls too hard, the roller will slip. Find the value of F at which this occurs.

  12. A yo-yo of Mass M, moment of inertia I, and inner and outer radii r and R, is gently pulled by a string with tension T as shown in the diagram below. The coefficients of friction between the yo-yo and the table are μs and μk.
    (a) Find the acceleration.
    (b) Find the friction acting on the yo-yo.
    (c) At what value of T will the yo-yo begin to slip?

    Slipping Objects

  13. A bowling ball of Radius R is given an initial velocity of v0 down the lane and a forward spin of w0 = 3v0/R. It first slips when it makes contact with the lane, but will eventually start to roll without slipping. The coefficient of kinetic friction is µk.
    (a) What is the direction of the frictional force? Explain.
    (b) For how long does the ball slide before it begins to roll without slipping?
    (c) What is the speed of the bowling ball when it begins to roll without slipping?
    (d) What distance does the ball slide down the lane before it starts rolling without slipping?

  14. A solid sphere is sliding (not rolling!) across a frictionless surface with speed v0. It slides onto a surface where the coefficient of kinetic friction is µ. Eventually it will start to roll without slipping.
    (a) What is the direction of the frictional force? Explain.
    (b) For how long does the sphere slide before it begins to roll without slipping?
    (c) What is the speed of the sphere when it begins to roll without slipping?
    (d) What distance does the sphere slide it starts rolling without slipping?

  15. A ball is placed on an incline as shown in the diagram below. The upper part of the incline is frictionless, so the ball slides but does NOT rotate. At point A, when its speed is 4.50 m/s, it reaches a rough portion of the incline where µk = 0.20. Here the ball starts to slip.
    (a) How long does it take for the ball to roll without slipping?
    (b) How far down the incline from point A does this occur?
    (c) What is the speed of the ball when it starts to roll without slipping?

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