## Unit – I B |

## Derivations of Relations

### Derivation of Relation Between Linear Velocity and Angular Velocity:

- Consider a particle performing uniform circular motion, along a circumference of circle of radius ‘r’ with constant linear velocity ‘v’ and constant angular speed ‘ω’ moving in anticlockwise direction as shown in diagram.
- In small interval of time ‘δt’, suppose particle moves from P to Q through a distance ‘δx’ and subtends the angle ‘δθ’ at the centre O of circle. By geometry δx = r . δθ
- If time interval is very very small then arc PQ can be considered to be almost a straight line.
- Therefore magnitude of linear velocity is given by

- Thus the linear velocity of a particle performing uniform circular motion is radius times its angular velocity.
- In vector form above equation can be written as i.e. The linear velocity can be expressed as vector product of angular velocity and radius vector.

**Proof of **

- For smaller magnitudes angular displacement, angular velocity are vector quantities. Let ( r) be the position vector of the particle at some instant. Let the angular displacement in small time δt be . Let ( dq) the corresponding linear displacement (arc length) be ( ds)
- By geometry
- Dividing both sides of equation by δt and taking limit

**Derivation of Expression for Period of Revolution:**

- Let us consider particle performing uniform circulation motion. Let ‘T’ be its period of revolution. During periodic time (T), particle covers a distance equal to the circumference 2pr of circle with linear velocity v.

This is an expression for the period of revolution for particle performing uniform circular motion.

**Derivation of Expression for Angular velocity of a Body Performing Circular Motion:**

The angular velocity of a body performing uniform circular motion is given by

In one period i.e. in time T seconds, the body performing uniform circular motion traces an angle of 2p radians..

Where ‘n’ is the frequency of U.C.M. and ‘N’ is angular speed of the body in r.p.m.

**Derivation of Expression for Angular Acceleration of a Body Performing Circular Motion:**

- When body is performing non uniform circular motion, its angular velocity changes. Hence the body possesses angular acceleration.

The rate of change of angular velocity w.r.t. time is called as the angular acceleration.

- We know that acceleration is the rate of change of velocity with respect to time.

- Where ω = angular velocity of the particle performing UCM. Now radius of circular path is constant.

This is the relation between the angular acceleration and linear acceleration in uniform circular motion.

- Where ‘a’ is angular acceleration. Hence, linear acceleration = radius x angular acceleration.
- If speed is increasing linear acceleration is in the same direction as that of linear velocity. If speed is decreasing linear acceleration is in the opposite direction to that of linear velocity. It is also referred as tangential acceleration.
- For uniform circular motion α = 0.

### Notes:

- For uniform circular motion Angular speed ω = constant, Linear speed v = constant, kinetic energy = constant, Angular momentum (L) = constant, angular acceleration α = 0 and the tangential acceleration a
_{T}= 0. - When a body is performing a uniform circular motion with velocity ‘v’ in a circle of radius ‘r’, then in half the revolution its velocity changes by ‘2v’.

- When a body is performing a uniform circular motion with velocity ‘v’ in a circle of radius ‘r’, then in half the revolution its displacement is ‘2r’ and distance travelled is πr.
- The angle between the radius vector and linear velocity of a particle performing circular motion is 90O. Hence their scalar product is zero.
- When a stone is tied to a string and rotated in a circle with a constant speed v, If the string is released then it flies along the tangent at a point of release on the circular path.
- If a graph is drawn by plotting tangential velocities of particles of a rigid body on y – axis and their distances from the axis of rotation on the x – axis, then the graph will be a straight line passing through the origin with a slope w.
- In uniform circular motion the magnitude of linear velocity and magnitude of linear (centripetal) acceleration remains constant but their direction changes continuously.