Question

A ball enters a frictionless track spiraling upward with a speed vo. The radius of the track is half a meter and it has a pitch (ɵ) of 15°. (a) If the initial speed of the ball is 12 m/s, how high vertically will it rise. (b) Compare your answer to part (a) with the height the ball would rise if it had been thrown vertically upward, into free-fall, with the same initial speed. Explain your results. (c) If you wish the ball to make exactly 5 revolutions before momentarily coming to rest, how fast must it initially be moving when entering the spiral track? This must be solved with algebra, not calculus.

Answer 1

(a)

Given

The initial speed of the ball = 12 m/s

The initial kinetic energy of the ball is

For the maximum height, the whole of its kinetic energy will convert into potential energy. therefore

We know that U = mgh

Considering the acceleration due to gravity

(b)

If the ball is thrown vertically upward then the same approach can be used to solve this problem.

We have to find the displacement of the ball when it comes to rest.

here final velocity of the ball at the top is zero. i.e. v = 0

(c)

The radius of the ball R = 0.5 m

The distance covered by the ball in 5 revolutions is

But this displacement is at 15 degrees so the net vertical displacement is

The velocity required to attain this height can be calculated by using the formula