## Unit – I A |

## Introduction to Circular Motion:

**Circular Motion:**

- The motion of a particle along the circumference of a circle is called as circular motion.

#### Examples:

- The motion of the earth around the sun.
- The motion of a satellite around the planet.
- The motion of an electron around the nucleus.
- The motion of a tip of blade of a fan

Thus, a Circular motion is a translational motion along a curved path.

**Radius Vector:( r)**

- A vector drawn from the centre of a circular path to the position of the particle at any instant is called a radius vector at that instant. Radius vector is also called as a position vector.
- In the figure at position P, r or OP is a position vector.
- The magnitude of the position vector is equal to the radius of the circular path.
- For circular motion the magnitude of the radius vector is constant but its direction changes continuously.

**Instantaneous Velocity:(****v****)**

- A linear velocity of a particle performing a circular motion, which is directed along the tangent to the circular path at given point on the circular path at that instant is called instantaneous velocity. Instantaneous velocity is also called as tangential velocity.
- For uniform circular motion the magnitude of instantaneous velocity is always constant but direction changes continuously. For non uniform circular motion the magnitude and direction of instantaneous velocity changes continuously.
- The tangential velocity is directed perpendicular to the direction of radius vector.

**Uniform Circular Motion:**

- The motion of a particle along the circumference of a circle with a constant speed is called uniform circular motion (U.C.M.).

e.g.

- The motion of the earth around the sun.
- Motion of an electron around the nucleus.

** Axis of Rotation:**

- The normal drawn to the plane of the circular path through the centre of the circular path is called the axis of rotation.

**Angular Displacement:**

- For a particle performing a circular motion the angle, traced by the radius vector at the centre of the circular path in a given time is called the angular displacement of the particle at that time.
- It is denoted by ‘θ’.
- Its S.I. unit is radian (rad).
- It is dimensionless quantity. [MºLºTº]

**Direction of Angular Displacement:**

For smaller magnitude (infinitesimal) angular displacement is a vector quantity and its direction is given by the right hand thumb rule.

**Statement:**

“If we curl the fingers of our right hand and hold the axis of rotation with fingers pointing in the direction of motion then the outstretched thumb gives the direction of the angular displacement vector”.

**Sign Convention:**

An angular displacement in counter clock-wise direction is considered positive and that in clockwise direction is considered as negative.

Vector relation between linear and angular displacement is

**Angular Velocity:**

- The rate of change of angular displacement with respect to time is called as the angular velocity of the particle
- It is denoted by letter ‘ω’.
- Its S.I. unit is radians per second (rad s
^{-1}). - Its dimensions are [MºLºT
^{-1}]. - Mathematically,
- For uniform circular motion the magnitude of angular velocity is given by

Where,

ω = Angular speed

T = Period

N = Angular speed in r.p.m.

n = Angular speed in r.ps. or Hz.

For uniform circular motion

Where, θ = Angular displacement

t = time taken

### Direction of Angular Velocity:

For smaller magnitude (infinitesimal) the angular velocity is vector quantity. It’s direction is given by the right hand thumb rule.

**Statement:**

“If we curl the fingers of our right hand and hold the axis of rotation with fingers pointing in the direction of motion then the outstretched thumb gives the direction of the angular velocity vector”.

Thus, the direction of angular velocity is the same as that of angular displacement.

### Angular Speed:

The angle traced by radius vector in unit time is called the angular speed or The magnitude of angular velocity is known an angular speed.

**Note:**

Uniform motion is that motion in which both the magnitude and direction of velocity remains constant. In UCM the magnitude of velocity is constant but its direction changes continuously. Hence UCM is not uniform motion.

### Angular Acceleration:

- The average angular acceleration is defined as the time rate of change of angular velocity.
- It is denoted by letter ‘α’ .
- Its S.I. unit is radians per second square (rad /s2).
- Its dimensions are [MºLºT
^{-2}]. - Mathematically,
- If initial angular velocity of particle changes from initial angular velocity ω
_{1}to final ω_{2}angular velocity in time ‘t’ then

**Direction of Angular Acceleration:**

- Direction of angular acceleration is given by right hand thumb rule.
- If the angular velocity is increasing then the angular acceleration has the same direction as that of the angular velocity.
- If the angular velocity is decreasing then the angular acceleration has the opposite direction as that of the angular velocity.

### Right Handed Screw Rule:

- When right handed screw is rotated in the sense of revolution of the particle, then the direction of the advance of the screw gives the direction of angular displacement vector.

### Period of Revolution:

- The time taken by a particle performing uniform circular motion to complete one revolution is called as period of revolution or periodic time or simply period (T).
- It is denoted by ‘T’ .
- The S. I. Unit of period is second (s).
- Its dimensions are[MºLºT
^{1}].

### Frequency of Revolution:

- The number of revolutions by the particle performing uniform circular motion in unit time is called as frequency (n) of revolution.
- The frequency is denoted by letter (n).
- The S. I. Unit of frequency is hertz (Hz).
- Its dimensions are [MºLºT
^{-1}]. - In time T the particles complete one revolution. Thus the particle completes 1/T revolutions in unit time. Thus n = 1/T.

### Notes:

- The second hand of a clock takes 60 seconds to complete one rotation. Its angular speed is 0.105 rad /s.
- The minute hand of a clock takes 60 minutes = 60 x 60 seconds to complete one rotation. Its angular speed is 1.746 x 10
^{-3}rad /s. - The hour hand of a clock takes 12 hours = 12 x 60 x 60 seconds to complete one rotation.. Its angular speed is 1.455 x 10
^{-4}rad /s. - The earth takes 24 hours to complete one rotation about its axis. The angular speed of the earth of its rotation about its axis is 7.273 x 10
^{-5}rad /s. - The ratio of angular speeds of a second hand of a clock and a minute hand of a clock is 60:1.
- The ratio of angular speeds of a minute hand of a clock and an hour hand of a clock is 12:1.
- The ratio of angular speeds of a second hand of a clock and a the hour hand of a clock is 720:1.