## Circular motion

If a particle is moving along a circular path then the particle is said to be in Circular motion .

If the speed of the particle is constant then it is called Uniform circular motion.

In case of circular motion, radial and tangential directions are important. In case of circle normal direction coincides with radially inward direction.

The direction of velocity at any point is given by the tangent drawn at that point to the circular path.

If the speed of the particle is constant then it is called Uniform circular motion.

In case of circular motion, radial and tangential directions are important. In case of circle normal direction coincides with radially inward direction.

The direction of velocity at any point is given by the tangent drawn at that point to the circular path.

From the figure it is clear that the direction of velocity of particle changes continuously. i.e., the velocity of the particle is changing continuously. Hence the particle must possess some acceleration.

This acceleration lies along the radius towards the center at any instant. Hence it is called Centripetal acceleration.

From the definition of acceleration and with the help of calculus it can be proved that centripetal acceletration for a particle moving with a speed 'v' along a circular path of radius 'r' is

This acceleration lies along the radius towards the center at any instant. Hence it is called Centripetal acceleration.

From the definition of acceleration and with the help of calculus it can be proved that centripetal acceletration for a particle moving with a speed 'v' along a circular path of radius 'r' is

Note :

1. For a curve, the direction of centripetal acceleration is along the normal towards the center of curvature.

2. From the figure, the direction of acceleration is changing continuously. That is acceleration is non uniform even though speed is constant.

1. For a curve, the direction of centripetal acceleration is along the normal towards the center of curvature.

2. From the figure, the direction of acceleration is changing continuously. That is acceleration is non uniform even though speed is constant.

3. Acceleration of a particle having uniform circular motion is not constant.

Centripetal force :

A particle moving in a circle is accelerated and acceleration can be produced in an inertial frame only if a resultant force acts on it. The resultant force along the normal direction, directed towards the center is called Centripetal force.

A particle moving in a circle is accelerated and acceleration can be produced in an inertial frame only if a resultant force acts on it. The resultant force along the normal direction, directed towards the center is called Centripetal force.

Linear kinematic quantities (translational) :

Translatory motion can be analysed either with cartesian or with polar coordinate system.

The linear kinematic quantities are distances (s), displacement ( ), speed (v), velocity ( ), acceleration ( ), etc.,

Angular kinematic quantities :

They can be analysed with polar coordinate system.

Translatory motion can be analysed either with cartesian or with polar coordinate system.

The linear kinematic quantities are distances (s), displacement ( ), speed (v), velocity ( ), acceleration ( ), etc.,

Angular kinematic quantities :

They can be analysed with polar coordinate system.

Angular displacement :

Change in angular position is called Angular displacement.

Change in angular position is called Angular displacement.

Note : During one complete revolution the particle makes an angular displacement of 2π radians.

Average angular velocity :

Angular displacement per unit time is called Average angular velocity.

Angular displacement per unit time is called Average angular velocity.

Instantaneous angular velocity : Angular velocity at a particular instant is given by instantaneous angular velocity.

Note :

1. Most often angular velocity is also measured in revolutions per minute (rpm) and also in revolutions per second (rps).

1. Most often angular velocity is also measured in revolutions per minute (rpm) and also in revolutions per second (rps).

2. The time taken by a particle to complete one revolution along its circular path is called Time period.

Angular acceleration : Change in angular velocity per unit time is called Angular acceleration

Note :

Relation between linear and angular velocities :

If a particle covers a distance 's' along a circular path which subtends an angle 'θ' at the center of the circle then distance travelled 's' can be expressed as

If a particle covers a distance 's' along a circular path which subtends an angle 'θ' at the center of the circle then distance travelled 's' can be expressed as

Tangential acceleration of particle :

Rate of change of speed is given by

Rate of change of speed is given by

If the particle is in uniform circular motion, its angular velocity ω is constant and angular acceleration will be zero.

Circular turning and Banking of roads :

When vehicles go through turnings, they travel along a nearly circular arc. There must be some force which will produce the required acceleration. If the vehicle goes in a horizontal circular path, this resultant force is also horizontal. Consider the situation as shown in figure given below. A vehicle of mass M moving at a speed 'v' is making a turn on the circular path on radius 'r'. The external forces acting on the vehicle are

When vehicles go through turnings, they travel along a nearly circular arc. There must be some force which will produce the required acceleration. If the vehicle goes in a horizontal circular path, this resultant force is also horizontal. Consider the situation as shown in figure given below. A vehicle of mass M moving at a speed 'v' is making a turn on the circular path on radius 'r'. The external forces acting on the vehicle are

- Weight Mg
- Normal contact force N and
- friction f(s).

If the road is horizontal, the normal force N is vertically upward. The only horizontal force that can act towards the center is the friction f(s). This is static friction and is self adjustable. The tyres get a tendency to skid outward and the frictional force which opposes this skidding acts towards the center. Thus, for a safe turn we must have

Friction is not always reliable at circular turns if high speeds and sharp turns are involved. To dependence on friction, the roads are banked at the turn so that the outer part of the road are banked at the turn so that the outer part of the road is somewhat is lifted up as compared to the inner part.

The surface of the road makes and angle θ with the horizontal throughout the turn. The normal force N makes an angle of θ with the vertical. At the correct speed, the horizontal component of N is sufficient to produce the acceleration towards the center and the self adjustable frictional force keeps its value zero. Applying Newton's second law along the radius and the first law in vertical direction,

The angle θ depends on the speed of the vehicle as well as on the radius of the turn. Roads are banked for the average expected speed of the vehicles. If the speed of a particular vehicle is a little less or little more than the correct speed, the self adjustable static friction operates between the tyres and the road and the vehicle does not skid of slip. If the speed is too different from the above equation, given the maximum friction can not prevent a skid or slip.

Centrifugal force :

It is the pseudo force observed by a person in a frame in circular motion. Centrifugal force should be used only in non-inertial frame. In an inertial frame we don't use this force.

Centrifugal force :

It is the pseudo force observed by a person in a frame in circular motion. Centrifugal force should be used only in non-inertial frame. In an inertial frame we don't use this force.