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Understanding Angular Displacement and Velocity in Circular Motion

This section explains the concepts of angular displacement and angular velocity in rotating objects. Angular displacement is the angle through which an object moves on a circular path and is expressed in radians. It is calculated using the formula for revolutions completed. Angular velocity measures how fast an object rotates over time and is also expressed in radians per second. Linear velocity, on the other hand, relates to the distance traveled along the arc per time unit. These concepts are crucial for understanding circular motion in physics.

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Understanding Angular Displacement and Velocity in Circular Motion

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  1. Section 6-2 Linear and Angular Velocity

  2. Angular displacement – As any circular object rotates counterclockwise about its center, an object at the edge moves through an angle relative to its starting position known as the angle of rotation.

  3. Determine the angular displacement in radians of 4.5 revolutions. Round to the nearest tenth. • Note – Each revolution equals 2π radians. • For 4.5 revolutions, the number of radians is = 28.3 radians

  4. Determine the angular displacement in radians of 8.7 revolutions. Round to the nearest tenth. • 8.7 x 2π=54.7 radians

  5. Angular velocity – the change in the central angle with respect to time as an object moves along a circular path. If an object moves along a circle during a time of t units, then the angular velocity, w, is given by Where θ is the angular displacement in radians.

  6. Determine the angular velocity if 7.3 revolutions are completed in 5 seconds. Round to the nearest tenth. • First calculate the angular displacement • 7.3 x 2π = 45.9 • w=45.9/5 = 9.2 radians per second

  7. Determine the angular velocity if 5.8 revolutions are completed in 9 seconds. Round to the nearest tenth. • 4.0 radians/s

  8. Angular velocity is the change in the angle with respect to time. • Linear velocity is the movement along the arc with respect to time.

  9. Linear Velocity • Linear velocity – distance traveled per unit of time • If an object moves along a circle of radius of r units, then its linear velocity v is given by • Where θ is the angular displacement therefore v=rw

  10. Determine the linear velocity of a point rotating at an angular velocity of 17π radians per second at a distance of 5 centimeters from the center of the rotating object. Round to the nearest tenth.

  11. Determine the linear velocity of a point rotating at an angular velocity of 31π radians per second at a distance of 15 centimeters from the center of the rotating object. Round to the nearest tenth. 1460.8 cm/s

  12. Pg 355

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