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Circular Motion Dynamics


A small sphere of mass \( m \) is moving on the inner surface of a large hemispherical bowl of radius \( R ,\) along a horizontal circle equidistant from the center of the bowl \(O.\) As shown in the above diagram, the distance from \(O\) to a point \(P\) on the circle is \(R\) and the distance between \(O\) and the center of the circle \(C\) is \(\lvert\overline{OC}\rvert = \frac{R}{2}. \) What is the force exerted by the sphere on the bowl?

Take gravitational acceleration as \( g. \)

A disc of radius \( r = 50 \text{ cm} \) is rotating about its axis with an angular speed of \( 40 \text{ rad/s}. \) It is gently placed on a perfectly frictionless horizontal surface like in the figure above. Find the linear speed of point \(A.\)

Old time mills for grinding wheat or corn into flour were often powered by a water wheel. Some water wheels had water falling over paddles, other water wheels dipped their paddles in a moving river, which as it flowed past turned the wheel. One particular mill has a water wheel with radius \(3~\mbox{m}\) suspended above a river such that the bottom edge of the paddles are just in contact with and move with the river water. If the river flows at \(0.5~\mbox{m/s}\) then what is the period of the spinning water wheel?

A roller coaster works by gravitational energy. The coaster car is pulled up to a high point and then released, rolling downwards on the track through all manners of curves and loops. I have a short roller coaster car that I pull up to the top of a hill of height \(H\). The coaster car is released from this height and must go around a perfectly circular vertical loop with a radius of 20 meters (and the bottom of the loop is on the ground). If I don't want the coaster car to fall off the loop at any point, what should be the minimum value of \(H\) in meters?

Details and assumptions

  • Neglect friction and air resistance.
  • Treat the roller coaster car as a point.

What is the maximum speed with which a \( 1600 \text{ kg} \) car can make a left turn around a curve of radius \( 49 \text{ m} \) on a level (unbanked) road without sliding?

The static friction constant between the car and the road is \( 0.1, \) and the gravitational acceleration is \( g = 10 \text{ m/s}^2. \)


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