Uniform Circular Motion

Acceleration When an object moves at a constant speed in a circular path, it is constantly changing direction – accelerating Δv

Centripetal Acceleration Δv always points to the center of the circle Since a = Δv/t, a must also be pointed to the center of the circle Centripetal Acceleration – center seeking acceleration Also called Radial Acceleration – directed along the radius Symbol - a r

Equation a r = v 2 / r v is the velocity and r the radius Greater v, faster direction changes, greater the a r Greater r, less rapidly changing direction, lower the a r

v and a are always perpendicular to one another at any given point on a circle v a

Frequency and Period Circular motion is often described in terms of frequency (f) and period (T) Frequency – revolutions per sec – unit Hz Period – time for one revolution – unit sec The velocity is the distance traveled per time

Distance is the circumference of the circle which is 2π r v = 2π r / T

Example 1 A 150 g ball at the end of a string is revolving in a horizontal circle of radius.600 m. The ball makes 2.00 revolutions in a second. What is the centripetal acceleration?

Example 2 The moon’s nearly circular orbit about the earth has a radius of about km and a period of 27.3 days. Determine the acceleration of the moon toward the earth.

Force Objects accelerating must have a force acting on them therefore there must be a force keeping an object in a circular path This force is called Centripetal Force (F r ) F r = ma r = mv 2 / r

Since a r is directed into the center of the circle, so is F r This force is always applied by other objects

Factious Force There is no force pulling objects out from the center of a circle There is an illusion of an outward force caused by inertia This factious force is called centrifugal force

Example 3 What force must a person exert on a string attached to a.150 kg ball in order to keep it revolving in a horizontal circle of radius.600 m ? The ball makes 2.00 revolutions per second.

Example 4 In a tetherball game a.85 kg ball is hit in a horizontal circle around a pole. The rope makes an angle of 35° with the pole and is holding the ball with a tension of 12.5 N. The radius of its circular path is 1.25 m. Find the centripetal acceleration, centripetal force, the velocity, and period.