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Circular Motion

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.

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Circular Motion

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  1. Circular Motion

  2. 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?

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

  4. Circular Motion • Circumference (distance around) = 2r • So 2 radians = one full circle. • So  radians = 180º • To convert rad to deg: • deg = rad x 180/ • 1 rad = 57.3º • To convert deg to rad: • rad = deg x /180 • 1º = 0.0175 rad

  5. Common Angles

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

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

  8. 12.0 m  r = 5.00 m 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

  9. t 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) = 0.105 rad/s

  10. 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.

  11. 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 = ?

  12. 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

  13. 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 = ?

  14. 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

  15. 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.

  16. v2 = 2r r Centripetal Acceleration • Centripetal Acceleration (ac) – causes a change in direction. • Perpendicular to direction of motion. • Measured in m/s2. • ac =

  17. Centripetal Acceleration

  18. 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/s2? • Given: • r = 3.50 m • ac = 20.0 m/s2 • Want: •  (in rpm) = ?

  19. Centripetal Acceleration • ac = 2r • 20.0 m/s2 = 2(3.50 m) • 2 = 5.71 /s2 •  = 2.39 rad/s • Now convert to rpm: • Recall that 1 rpm = 0.105 rad/s • (2.39 rad/s) / 0.105 = 22.8 rpm

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