Rotational Dynamics

Problems

practice

1. A kind of Atwood's machine is built from two cylinders of mass m₁ and m₂; a cylindrical pulley of mass m₃ and radius r; a light, frictionless axle; and a piece of light, unstretchable string. The heavier mass m₁ is held above the ground a height h and then relased from rest.
   1. Draw a free body diagram showing all the forces acting on…
      1. the heavier mass
      2. the lighter mass
      3. the pulley
   2. Write the equation stating Newton's second law of translational motion for…
      1. the heavier mass
      2. the lighter mass
      and rotational motion for…
      3. the pulley
   3. Determine the translational acceleration of…
      1. the heavier mass
      2. the lighter mass
      and the rotational acceleration of…
      3. the pulley
   4. Determine the tension in the side of the string connected to
      1. the heavier mass
      2. the lighter mass
      and the upward force of the axle on…
      3. the pulley
   5. Lastly, determine…
      1. the time it takes for the heavier mass to reach the ground
      2. its speed on impact
      3. the rotational speed of the pulley at this time

2. 

   A roll of toilet paper is held by the first piece and allowed to unfurl as shown in the diagram to the right. The roll has an outer radius R = 6.0 cm, an inner radius r = 1.8 cm, a mass m = 200 g, and falls a distance s = 3.0 m. Assuming the outer diameter of the roll does not change significantly during the fall, determine…
   1. the tension in the sheets
   2. the translational acceleration of the roll
   3. the angular acceleration of the roll
   4. the final translational speed of roll
   5. the final angular speed of the roll
3. The top shown below consists of a cylindrical spindle of negligible mass attached to a conical base of mass m = 0.50 kg. The radius of the spindle is r = 1.2 cm and the radius of the cone is R = 10 cm. A string is wound around the spindle. The top is thrown forward with an initial speed of v₀ = 10 m/s while at the same time the string is yanked backward. The top moves forward a distance s = 2.5 m, then stops and spins in place.

   Magnify

   Using rotational dynamics (and kinematics) determine…

   1. the moment of inertia I of the top (essentially, the moment of inertia of a cone)
   2. the tension T in the string
   3. the final angular velocity ω of the top
   4. the length ℓ of string wound around the spindle
4. Write something different.

conceptual

1. Describe a test involving rotational principles that can be used to distinguish raw eggs from hard boiled eggs.
2. Some airplanes have only one propeller, but all helicopters have two rotors. (A helicopter with only one rotor is unflyable.)
   1. Why must helicopters always have two rotors?
   2. How does a single propeller aircraft manage to avoid the problems encountered by a single rotor helicopter?
3. The Physics Teacher has published several articles containing free body diagram worksheets. They are available free to members of the American Association of Physics Teachers (AAPT). Everyone else has to pay.
   • Free-body diagrams revisited—II. James E. Court. The Physics Teacher. Vol 37 No. 8 (1999): 490–495.
     • RE1–RE16: Rotational Equilibrium
     • RN1–RN9: Rotational Nonequilibrium

algebraic

1. A simple catapult is composed of the following parts, assembled and momentarily held in place as shown in the diagram below…
   • a beam of mass m, length 2r, and moment of inertia about its center of mass ⅓mr²
   • a compact projectile of mass ¼m
   • a compact counterweight of mass 4m
   • a triangular base

   

   Determine the following quantities in terms m, r, and g at the instant the counterweight is released…
   1. the moment of inertia of the catapult about the pivot
   2. the net torque on the catapult
   3. the tangential acceleration of the projectile.