# Homework Help: Time Period in Vertical Circular Motion

1. Mar 25, 2010

### shreyasrulez

1. The problem statement, all variables and given/known data

A bucket is rotated in a vertical circle such that the water just does not fall. Radius of the circle is R. Gravitational acceleration: g. Find the Time period of one circular motion.

http://img148.imageshack.us/img148/6583/bucket.jpg [Broken]

3. The attempt at a solution

I obtained an expression for w (omega) (i.e. d(theta)/dt) in terms of theta and integrated. I want to know whether this is right and if there is a better approach (possibly not involving integration).

For the water to just not fall, the velocity at the lowest point will be $$\sqrt{5gR}$$

Now here is the energy conservation equation for a general point,

$$1/2m*5gR = 1/2mv^{2} + mgR(1 - cos\theta)$$

Solving for v,

$$v = \sqrt{(3 + 2 cos\theta) gR}$$
$$\omega = d\theta/dt = \sqrt{(3 + 2 cos\theta) g/R}$$

Now separating the variables and integrating,

$$T (Time Period) = \int d\theta/(\sqrt{(3 + 2 cos\theta) g/R})$$ (using limits 0 to 2 pi)

Here I'm having trouble integrating this thing, so I'm wondering whether there is a simpler method.

Last edited by a moderator: May 4, 2017