How do I find the Centripetal Force?

Question

A small ball of mass 24 kg is suspended from a string of length 63 cm, and whirled in a circle lying in the horizonal plane. The acceleration of gravity is 9.8 m/s^2. If the string makes an angle of 36 with the vertical, find the centripetal force experience by the ball.

Answer 1

If you drew a diagram showing the ball frozen for an instant, it would have only two forces acting on it - gravity pulling straight down, and the tension of the string at a 36° angle with the vertical.

The string's tension can be broken into x and y components. Tx = T sin 36 and Ty = T cos 36.

The ball isn't accelerating up or down, so the gravitational force must be equal to the Ty force.

So mg = T cos 36
24 kg x 9.8 m/s^2 = T (.809)
T = 291 N

The centripetal force is simply the force directed toward the center of the circle the ball is traveling in. In this case it is Tx.

Tx = T sin 36
Tx = 291 N (.588) = 171 N

So, the centripetal force is 171 N.

By the way, a 24 kg ball isn't small! It's heavier than 3 bowling balls!

Answer 2

Tsin36 = 24v^2/(.63sin36)

is the equation you get when you consider the horizontal components of the forces and the resulting acceleration.

Tcos36 = 24(9.8)

is the equation you get when you consider the y components of the forces acting on the mass.

Solve the system of equations for T and v, then you can calculate the centripetal force, it is either mv^2/(.63sin36) or
Tcos36.