Circular Motion Dynamics

         

A small sphere of mass m m is moving on the inner surface of a large hemispherical bowl of radius R, R , along a horizontal circle equidistant from the center of the bowl O.O. As shown in the above diagram, the distance from OO to a point PP on the circle is RR and the distance between OO and the center of the circle CC is OC=R2.\lvert\overline{OC}\rvert = \frac{R}{2}. What is the force exerted by the sphere on the bowl?

Take gravitational acceleration as g. g.

A disc of radius r=50 cm r = 50 \text{ cm} is rotating about its axis with an angular speed of 40 rad/s. 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.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 m3~\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 m/s0.5~\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 HH. 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 HH 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 kg 1600 \text{ kg} car can make a left turn around a curve of radius 49 m 49 \text{ m} on a level (unbanked) road without sliding?

The static friction constant between the car and the road is 0.1, 0.1, and the gravitational acceleration is g=10 m/s2. g = 10 \text{ m/s}^2.

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