Topics are arranged as per SYLLABI published by Maharashtra State Board of Secondary and Higher Secondary Education.

Circular motion of a particle is defined as a motion of particle along a circular path (complete or partial arc of a circle).

**Angular Displacement: ** It is defined as the angle turned by the particle from some reference line. Angular displacement is usually measured in radians.

Finite angular displacement is a scalar but an infinitesimally small displacement is a vector.

**Angular Velocity:** It is defined as the rate of change of the angular displacement of the body.

Angular Velocity =

It is an axial vector whose direction is given by the right hand rule. Its unit is rad/s

**Angular Acceleration:** It is the rate of change of angular velocity.

Thus It"s Unit is

**Velocity:** A particle in a circular motion has two types of velocities and corresponding two speeds.

Linear Velocity or speed (v): and

Angular velocity or speed (): and

Relation between linear speed (v) and angular speed () is .

Circular motion performed with a constant speed is known as uniform circular motion. Total time taken by the particle performing uniform circular motion to complete one full circular path is known as time period. where is angular velocity.

When a body performs uniform circular motion its speed remains constant but velocity continuously changes due to change of direction. Hence a body is continuously accelerated and the acceleration experienced by the body is known as centripetal acceleration.

All objects in uniform circular motion must experience some form of uniform centripetal acceleration.

**Acceleration:** Acceleration of a particle in circular motion has two components :

**Tangential Acceleration:**a_{t}which is the component of acceleration in the direction of velocity. It is denoted as**Radial Acceleration:**a_{r}is the component of acceleration towards the centre of the circular motion. This is responsible for a change in the direction of velocity. It is denoted as

Centripetal Force | Centrifugal Force | |
---|---|---|

Required to move body along a circular path with a constant speed. | Virtual force due to incorporated effects of inertia. | |

Never acts by itself | Pseudo force (NOT Real) | |

Direction | Along the radius, acting towards the centre of the circle, on which the given body is moving. | Radially outwards, in a direction opposite to that of the centripetal force. |

Applications | Motion of a vehicle on a level circular road. Banking of a curved road Bending of a cyclist Motion of cyclist in a Death Well Motion along a vertical circle. | The centrifugal force cannot balance the centripetal force because they act on different bodies. It always act on the centre, directed away from centre. |

Some common examples of centripetal force:

Situation | Centripetal Force |
---|---|

Earth in orbit around the sun. | Gravitational force exerted by the sun. |

Vehicle taking turn on a level road on tyres. | Frictional force exerted by the road. |

Particle tied to string and whirled in horizontal circle. | Tension in the string. |

Electron revolving around the nucleus in an atom. | Coulomb Attraction (by proton). |

When a car is moving along a curved road, it is performing circular motion. For circular motion it is not necessary that the car should complete a full circle. We can consider arc of circle also as a circular path.

We know that centripetal force is necessary for circular motion. Lets consider motion of car along circular path, centripetal force for uniform circular motion of the car can be provided in two ways:

- Frictional force between the tyres of the car and the road
- Banking of Road

The process of raising the outer edge of a road over the inner edge through a certain angle is known as banking of road. Few reasons stating need for Banking of Roads are:

- To provide necessary centripetal force for circular motion.
- To reduce wear and tear of tyres due to friction.
- To avoid skidding
- To avoid overturning of vehicles

Banking angle is given as: .

Kindly note: Banking angle is independent of the mass of the vehicle. The proper velocity or optimum velocity on a road banked by an angle with the horizontal is given by: Where r = radius of curvature of road; g = acceleration due to gravity; = coefficient of friction between road and tyres.

Let us consider an object of mass m tied to the end of an inextensible string and whirled in a vertical circle of radius r.

At highest point A, Let us assume velocity is v_{1} The forces acting on the object at highest point are:

- Tension acting in downward direction (T
_{1}) - Weight acting in downward direction
- The centripetal force acting on the object A is provided partly by weight and partly by tension in the string:
- At highest point object must have certain minimum velocity at point A so as to continue in circular in path. This velocity is called the critical velocity. It is given by equation:

At Lowest Point B, Let us assume velocity is v_{2}. The forces acting on the object at lowest point are:

- Tension acting in upward direction (T
_{2}) - Weight mg acting in downward direction
- Tension in the string is denoted by:
- At lowest point, object velocity is maximum and denoted by .

Highest Point (A) | Lowest Point (B) | Midway | |
---|---|---|---|

Velocity | |||

Acceleration | g | 5g | 3g |

Total Energy |

Note: Total energy is conserved in vertical circular motion, at any point it is represented as .

Term | Analogous to |
---|---|

Angular displacement () | Linear displacement (s) |

Angular velocity ( = initial; = after time t ) | Linear velocity (u=initial; v = after time t) |

Angular acceleration () | Linear acceleration (a) |

Let us consider a particle moving along a circle with an angular velocity . Suppose that the particle is subjected to a uniform angular acceleration , so that its angular velocity goes on changing with time. If is its angular velocity after time t and is its angular displacement in time t, then the three kinematical equations for uniformly accelerated circular motion can be stated as follows: ; ;

Note: are called pseudo vectors or axial vectors.

- Angles measured in clockwise are negative and counter clockwise are positive.
- = 1 revolution.
- Finite angular displacement is a scalar quantity because it does not obey the laws of vector addition. It is a scalar quantity.
- Instantaneous angular displacement is a vector quantity.
- Angular speed is a scalar quantity but angular velocity is a vector quantity but both have same units i.e. rad/s.
- When a particle moves in a circle with constant speed, its velocity is variable because of changing direction.
- Circular motion is a two-dimensional motion in which the linear velocity and linear acceleration vectors lie in the plane of the circle but the angular velocity and angular acceleration vectors are perpendicular to the plane of the circle.
- Centrifugal force is a fictitious force and holds good in a rotating frame of reference.
- Whenever a car is taking a horizontal turn, the normal reaction is at the inner wheel.
- If the force acting on a particle is always perpendicular to the velocity of the particle, then the path of the particle is a circle. The centripetal force is always perpendicular to the velocity of the particle.
- If a vehicle is moving on a curved road with speed greater than the speed limit, the reaction at the inner wheel disappears and it will leave the ground first.
- K.E. of a body moving in horizontal circle is same throughout the path, but K.E. of the body moving in vertical circle is different at different places.
- If circular motion of the object is uniform, the objects will posses only centripetal acceleration. In the non-uniform circular motion, magnitude of the centripetal acceleration remains constant whereas it"e;s direction changes continuously (directed towards centre).
- Although centrifugal force is equal and opposite to centripetal force, yet it is not the reaction of centripetal force because reaction cannot exists without action while centrifugal force can exist without centripetal force.