39 Practice Makes Perfect IV

You have probably seen a swing carousel at some point in your life.  You can see an example below (full video here).

Let’s work out how fast the carousel needs to spin.  To do so, we will assume the top circular bit stays horizontal, and the swings just hang vertically from a cable.  The cables are attached a distance from the rotation axis, and the cables have a length .  Find the angular velocity at which the ride must spin in order for the swings to “fly up” until they reach a maximum angle with respect to the vertical (see picture below).  The mass of a person sitting on the chair is .  If you get stuck with your drawing, you can use the slider to reveal mine.

A large ball of mass and radius is attached to a string of length by gluing one end of the string to the surface of the ball. The other end of the string is nailed to a wall, so that the ball hangs as shown.  Find the normal force with which the ball is pushing on the wall.  If you get stuck with your drawing, you can reveal mine with the slider.

Consider the “double ramp” shown below (don’t move the slider yet).  The block on the right has a large mass , while the block on the left has a smaller mass , so that the large mass starts going down the ramp. The static and kinetic frictional coefficients are identical, and we’ll denote both as .

A. Find the minimum mass required for the big block to start going down hill.  If you’re having trouble with your picture, use the slider to reveal mean.

B. Suppose is larger than the minimum mass you found in part A.  Find the acceleration of the boxes as they move.