Kinematics of uniform circular motion
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Define the radian.
A radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. There are 2π radians in a full circle.
What is angular displacement?
Angular displacement (θ) is the angle, in radians, through which an object has moved on a circular path. It is a vector quantity with direction perpendicular to the plane of rotation.
Define angular speed.
Angular speed (ω) is the rate of change of angular displacement. It is measured in radians per second (rad s⁻¹).
State the relationship between angular speed (ω) and period (T).
The angular speed is equal to 2π divided by the period: ω = 2π / T. This is derived from one full revolution (2π radians) in one period (T).
State the relationship between linear speed (v), radius (r), and angular speed (ω) for an object in uniform circular motion.
The linear speed is equal to the radius multiplied by the angular speed: v = rω. This means that for a given angular speed, objects at larger radii have greater linear speeds.
A particle moves in a circle of radius 0.2m with a period of 5s. What is its angular speed?
ω = 2π / T = 2π / 5 = 1.26 rad s⁻¹
A car travels around a circular track of radius 50m at a constant speed of 10 m/s. What is the car's angular speed?
Use v=rω and rearrange to get ω = v/r = 10/50 = 0.2 rad/s
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