Ask Me Help Desk - DynamicsII:Motion in a Plane

DynamicsII:Motion in a Plane

A small ball rolls around a horizontal circle at height y inside a frictionless hemispherical bowl of radius R.

a)Find an expression for the ball's angular velocity in terms of R, y, and g(gravity).

b)What is the minimum value of w(angular velocity) for which the ball can move in a circle?

c)What is w(angular velocity) in rpm(revolutions per minute) if R=20cm and the ball is halfway up?

I have had one mf view of this thing in 6 hours. Unknown, answer this please!

Okay, I just learned that in a book yesterday, otherwise, I wouldn't be able to answer this, lucky! ;)

First, we need to find the inclination of the surface on which the ball is rolling.

Make a sketch, and you get:

$\cos\theta = \frac{R-y}{R}$

Ok, now since there is no friction, this becomes easier.

The normal force of the ball with the surface is given by:

$N = mg \cos \theta = \frac{mg(R-y)}{R}$

The component which provides the centripetal force is given by:

$F_c = F sin\theta = \frac{mg(R-y)}{R}\sin\theta$

$F_c = mr\omega^2$

$m(Rsin\theta)\omega^2 = \frac{mg(R-y)}{R} sin\theta$

[The distance from the centre of rotation is R sin theta]

$\omega = \sqrt{\frac{g(R-y)}{R^2} }$

$\omega = \frac{\sqrt{g(R-y)}}{R}$

b) This occurs when height y is minimum.

c) y = R/2 (in metres)

Find omega in rad/s and convert to rpm.

Hey thanks again! That was a tough one!

You're welcome :)