Circular Motion

Questions for Consideration  How do we measure circular motion?  What is a radian?  What are the angular analogs of linear motion?  What is centripetal acceleration?

Circular Motion  How do we measure circular motion? Typically use radians. Angle subtended by an arc equal in length to the circle’s radius.

Circular Motion CCircumference (distance around) = 2  r So 2  radians = one full circle. So  radians = 180º TTo convert rad to deg: deg = rad x 180/  1 rad = 57.3º TTo convert deg to rad: rad = deg x  /180 1º = rad

Common Angles

Arc Length  Arc Length (s) Measured in meters along circumference of circle.

Angular Displacement  Angular Displacement (  ) Measured in radians. CCW rotation = +  CW rotation = -   s = r  ++ --

Circular Motion  A wheel with r = 5.00 m spins counterclockwise so that an ant resting on the top travels 12.0 meters along the rim. Through what angular displacement did the wheel rotate? s = r  12.0 m = (5.00 m)(  )  = 2.40 rad r = 5.00 m 12.0 m 

Angular Velocity  Angular velocity (  )  =  Expressed as rad/s.  Can also be given in terms of revolutions per unit time. revolutions per minute (rpm)  1 rpm = (2  rad) / (60 s) = rad/s  t

Tangential Velocity  Tangential velocity (v) – the instantaneous velocity of an object moving in a circular path. Imagine a bucket being swung around on a rope. The bucket has a tangential velocity that is perpendicular to the rope. If the rope breaks, the bucket’s tangential velocity will become its linear velocity.

Tangential Velocity  Formula for tangential velocity: v = r   A child is riding a merry-go-round that is rotating at 30 rpm. How fast is the child moving if she is 2.5 m from the center? Given:   = 30 rpm  r = 2.5 m Want:  v = ?

Tangential Velocity  First, convert rpm to rad/s: 30 rpm = (30 * 2  rad) / (60 s) = 3.14 rad/s  v = r  v = (2.5 m)(3.1 rad/s) = 7.8 m/s

Tangential Velocity  A satellite moves around the Earth in a circular orbit with r = 10,000 km. If the satellite takes 2.76 hours to complete one orbit, calculate the satellite’s angular and tangential velocities. Given:  r = 10,000 km  t = 2.76 hr Want:   = ?  v = ?

Tangential Velocity   =  / t  = (2  rad) / (2.76 hr)  = 2.28 rad/hr  v = r  v = (10,000 km)(2.28 rad/hr) v = 22,800 km/hr

Centripetal Acceleration  Can something accelerate but maintain a constant speed? Yes, if it changes direction.  Acceleration = change in velocity / time  Change in velocity = speed up, slow down, or change direction.

Centripetal Acceleration  Centripetal Acceleration (a c ) – causes a change in direction. Perpendicular to direction of motion. Measured in m/s 2.  a c = v2v2 r =  2 r

Centripetal Acceleration

 An amusement park ride spins riders around so fast that they are seemingly stuck to the walls. If the ride has a radius of 3.50 meters, what angular velocity (in rpm) is necessary to create a centripetal acceleration of 20.0 m/s 2 ? Given:  r = 3.50 m  a c = 20.0 m/s 2 Want:   (in rpm) = ?

Centripetal Acceleration  a c =  2 r 20.0 m/s 2 =  2 (3.50 m)  2 = 5.71 /s 2  = 2.39 rad/s  Now convert to rpm: Recall that 1 rpm = rad/s (2.39 rad/s) / = 22.8 rpm