#### Transcript Circular Motion

1 of 29 © Boardworks Ltd 2010 2 of 29 © Boardworks Ltd 2010 What are radians? 3 of 29 © Boardworks Ltd 2010 Converting from radians The circumference of a circle is 2πr. Since a radian is based on an arc with length r, there are 2π radians in a full circle. degrees radians 360 2π 180 π 90 π/2 r 1 rad r 2π To convert from degrees into radians, multiply by . 360 360 To convert from radians into degrees, multiply by . 2π 4 of 29 © Boardworks Ltd 2010 Degrees and radians in a circle 5 of 29 © Boardworks Ltd 2010 Consequences of using radians 6 of 29 © Boardworks Ltd 2010 7 of 29 © Boardworks Ltd 2010 Angular displacement and velocity Angular displacement and angular velocity, like linear displacement and linear velocity, are vector quantities, i.e. quantities with a direction. Displacement, x, is a distance with a direction. Linear velocity, v, is a speed with direction. It is the rate of change of displacement with time. Δx v= Δt Angular displacement, Δθ, is the angle moved through relative to a specific axis. Angular velocity, ω, is the rate of change of angular –1. The units of ω are rad s displacement with time. Δθ ω= Δt 8 of 29 © Boardworks Ltd 2010 Period of circular motion 9 of 29 © Boardworks Ltd 2010 An equation for v The distance covered by an object moving in a circle in one period is the circumference of the circle, 2πr. v distance speed = time therefore 10 of 29 r v= 2πr T © Boardworks Ltd 2010 An equation for ω The angle an object moving in a circle moves through in one period is 2π rad. 2π Δθ ω= Δt therefore 11 of 29 2π ω= T © Boardworks Ltd 2010 Equation linking v and ω 12 of 29 © Boardworks Ltd 2010 Frequency Frequency, f, is the number of rotations per second. i.e. Because number of rotations frequency = time taken time taken period = number of rotations 1 T= f and 1 f= T 2π Using ω = , this gives ω = 2πf T 13 of 29 © Boardworks Ltd 2010 Matching equations 14 of 29 © Boardworks Ltd 2010 Can you calculate v and ω? 15 of 29 © Boardworks Ltd 2010 16 of 29 © Boardworks Ltd 2010 Centripetal acceleration Any object moving in a circle has a changing velocity, even if it is moving at a constant speed, because its direction is changing. It is always moving along a tangent to the circle. v v v time Acceleration is the rate of change of velocity. Δv a= Δt Therefore any object moving in a circle is accelerating. This is called the centripetal acceleration. 17 of 29 © Boardworks Ltd 2010 Deriving equations for a 18 of 29 © Boardworks Ltd 2010 Centripetal force 19 of 29 © Boardworks Ltd 2010 F = ma and direction of acceleration According to Newton’s second law, force = mass × acceleration (F = ma). v Acceleration and force therefore act in the same direction as one another. a It can therefore now be seen that the direction of centripetal acceleration, a, is towards the centre of the circle. This is perpendicular to the direction of the velocity, v. 20 of 29 © Boardworks Ltd 2010 Deriving equations for F 21 of 29 © Boardworks Ltd 2010 Matching centripetal forces 22 of 29 © Boardworks Ltd 2010 Centripetal force and acceleration 23 of 29 © Boardworks Ltd 2010 More advanced calculations 24 of 29 © Boardworks Ltd 2010 Effect of m, v and r on F 25 of 29 © Boardworks Ltd 2010 26 of 29 © Boardworks Ltd 2010 Glossary 27 of 29 © Boardworks Ltd 2010 What’s the keyword? 28 of 29 © Boardworks Ltd 2010 Multiple-choice quiz 29 of 29 © Boardworks Ltd 2010