Set 8 Problem number 6

• Important Note on Notation

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s); Extension

• Figure(s)

Problem

An object of mass 1 kg is moving at 9.99 m/s on a circle of radius 4 meters.

• What is its centripetal acceleration and what is the centripetal force holding it in the circle?

Solution

• The acceleration is v ^ 2 /r = ( 9.99 m/s) ^ 2 / ( 4 m) = 24.95 meters per second per second toward the center of the circle.
• To accelerate 1 kg at this rate requires F = m * a = ( 1 kg)( 24.95 m/s/s) = 24.95 Newtons of force.

Generalized Solution

• The centripetal acceleration of an object moving along a circle of radius r at velocity v is a = v^2 / r; this acceleration is directed toward the center of the circle.
• If the object has mass m, the force required to keep the object in its circular path is F = m a = m v^2 / r; this force is also directed toward the center of the circle.

Explanation in terms of Figure(s); Extension

The figure below shows an object moving with constant speed v on a circle of radius r.

• The acceleration required to keep the object moving in the circle, as opposed to the straight line along which it would travel if it had no acceleration, is a = v^2 / r, and is directed toward the center of the circle.
• If the object has mass m, the force required to keep it on its circular path will therefore be F = m a = m v^2 / r. This force will be directed in the same direction as the acceleration, toward the center of the circle.

Figure(s)