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Angular Kinematics

If you shrunk to the size of an ant, and were forced to live on the second hand of a wristwatch, it would behoove you to understand movement in a circle. Prepare for the worst with Angular Kinematics.

Problem Solving

When a ceiling fan is switched off, its angular velocity reduces to half after it making 36 rotations. How many more rotations will it make before coming to rest?

Details and Assumptions

  • Angular deceleration of the fan is uniform.

A mass \( m = 2 \text{ kg} \) which is attached to a vertical rod by a string revolves around the rod on a frictionless horizontal surface. As the mass revolves, the string winds to the rod, causing the distance between the mass and rod to decrease. When the distance between the mass and the rod is \( r_i = 12 \text{ m}, \) the mass has a speed of \( v_i = 8 \text{ m/s}. \) What is the speed of the mass at the moment its distance from the rod is \(r_f=4\text{ m}?\)

A solid sphere of mass \( 5 \text{ kg} \) and radius \( 2 \text{ m} \) that is initially at rest, rolls (without slipping) down a \( 72\)-meter-long inclined plane. If the sphere reaches the bottom of the plane after \(6\) seconds, what is its angular speed at that moment?

Geostationary satellites orbit at a height of \(35,786~\mbox{km}\) above the Earth's surface and orbit with the same period as the Earth (so they always stay above the same spot, i.e. geostationary). If you stand on the Earth's surface what is the ratio of a geostationary satellite's speed to your speed?

Details and assumptions

  • Assume the earth is a perfect sphere of radius 6370 km.
  • We're asking about tangential speeds in this question (in m/s), not angular velocities.

At the playground, there is a merry-go-round initially at rest, whose moment of inertia is \( 1320 \text{ kg} \cdot \text{m}^2.\) A boy weighing \( 33 \text{ kg} \) stands on it, and his distance from the center of the merry-go-round is \( 8 \) meters. If the boy begins to run in a circular path (about the center of the merry-go-round) with a speed of \( 6.0 \text{ m/s} \) relative to the ground, what is the angular speed of the merry-go-round?

Ignore any friction other than that between the surface of the merry-go-round and the boy.

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