Circular Motion How do we work out the velocity of something which is moving at constant speed in a circle ? Answer: We use the simple formula: But in this case, the distance is the circumference of the circle = 2r And the time is the time taken to complete a full revolution, T So velocity of an object moving in a circle of radius r is given by the formula: Where T is the time taken for 1 complete rotation. Alternatively we can say: Where f is the frequency of rotation, or number of rotations per second.

Radians and angular speed in radians per second Using ‘radians’ to measure angles r Definition of a ‘radian’:  The angle subtended by 1 radius of arc on a circle r Circumference of circle = 2r So how many radians in a full circle ?  = 1 radian Each radian gives an arc of length r So there must be 2 of them to make a full circle  Conversion degrees to radians: 360 = 2 radians  radians = 180 Using angular velocity ‘’ (omega) (radians/second) to measure speed of rotation For Linear Position, Displacement = Velocity  Time x = vt  = t For Angular Position, Angle = Angular Velocity  Time 1 revolution/sec   = 2 radians/sec 0.5 revolution/sec   =  radians/sec 2 revolution/sec   = 4 radians/sec Questions: 1. Calculate the angular velocity in radians per second of the minute hand of a clock 2. A wheel has  = 10 rad/s. What is this in degrees per second, and in revolutions per second.

Linking angular speed  with time for 1 full rotation T We have the formula for angular position  = t or When an object completes a full revolution, the angle in radians which it rotates through is... 2 radians And the time taken for it to do this is... the time for 1 full rotation T v so v Also, remember or v Questions Page 23. 1. a. b. c.

2. a. b. (i) (ii) 3. a. b. (i) (ii)

4. a. b. (i) (ii)

Circular Motion How do we work out the velocity of something which is moving at constant speed in a circle ? But in this case, the distance is the circumference of the circle And the time is the time taken to complete a full revolution So velocity of an object moving in a circle of radius r is given by the formula: Where T is the time taken for 1 complete rotation. Alternatively we can say: Where f is the frequency of rotation, or number of rotations per second.

Radians and angular speed in radians per second Using ‘radians’ to measure angles Definition of a ‘radian’: The angle subtended by 1 radius of arc on a circle Circumference of circle So how many radians in a full circle ? Each radian gives an arc of length r So there must be of them to make a full circle  Conversion degrees to radians: Using angular velocity ‘’ (omega) (radians/second) to measure speed of rotation For Linear Position, Displacement = Velocity  Time x = vt For Angular Position, 1 revolution/sec   = 0.5 revolution/sec   = 2 revolution/sec   =

Linking angular speed  with time for 1 full rotation T We have the formula for angular position or When an object completes a full revolution, the angle in radians which it rotates through is... And the time taken for it to do this is... the time for 1 full rotation T so or Questions Page 25.