Circular Motion, and Conservation of Energy problem

[leighzer]

A ball of mass m is spun in a vertical circle having radius R. The ball has a speed v(o) at its highest point. Take zero potential energy at the lowest point, and use the angle theta measured with respect to the vertical.

(a) Derive an expression for the speed v at any time as a function of R, theta, v(o), and g.
(b) What minimum speed v(o) is required to keep the ball moving in a circle?


I found part (a) to be:

v = *square root*[ v(o)^2 + 2gRsin(theta) ]


I'm not totally sure if this is correct, and i don't know how to find the minimum value of v(o).


Is someone able to help?

 

[leighzer]

The only thing I can think of for part (b) is that v(o) must be greater than or equal to:

-*sqr rt*[ 2gRsin(theta) ]

in order to keep the value inside the square root of the first equation positive.

 

[kyle8921]

I can confirm that your expression for the speed v is correct. This is derived from the conservation of energy principle, which states that the total energy of a system remains constant. In this case, the total energy is the sum of kinetic energy and potential energy, which can be represented as the following equation:

E = 1/2 mv^2 + mgh

Where m is the mass of the ball, v is the speed, g is the acceleration due to gravity, and h is the height of the ball.

At the lowest point of the circle, the ball has the maximum potential energy (mgh) and zero kinetic energy (1/2 mv^2). As the ball moves up the circle, its potential energy decreases while its kinetic energy increases. At the highest point, the ball has zero potential energy and maximum kinetic energy.

Using this principle, we can equate the energy at the highest point to the energy at any point on the circle, which gives us the following equation:

1/2 mv^2 = mgh + 1/2 mv(o)^2

Solving for v, we get the expression you derived:

v = *square root*[ v(o)^2 + 2gRsin(theta) ]

To find the minimum speed v(o) required to keep the ball moving in a circle, we can use the fact that the ball must have a minimum centripetal force acting on it to maintain circular motion. This force is given by:

F = mv^2/R

Equating this force to the weight of the ball (mg), we can solve for the minimum speed v(o):

mv(o)^2/R = mg

v(o)^2 = gR

v(o) = *square root*(gR)

Therefore, the minimum speed required to keep the ball moving in a circle is *square root*(gR). Any lower speed would not provide enough centripetal force and the ball would fall from the circle.

I hope this helps clarify any confusion and provides a better understanding of circular motion and conservation of energy.