Circular Motion for IIT JEE

IIT JEE is considered to be one of the most competitive exams which lends the students to their dream colleges and dream jobs. In the journey of their preparation for IIT JEE students often get stuck trying to understand certain topics and concepts. To make the journey flawless for the students, a team of expert faculty at Vedantu has worked collectively to explain all the topics important for IIT JEE in the most lucid manner. The main goal of Vedantu is to help the students level up their preparation with the right guidance and support. 

What is Circular Motion?  

The motion of an object is said to be circular if it moves in a way that its distance from the center point always remains the same. When any object is roughly moving on a circular path it is called circular motion. The speed of the body during the motion can either change or maybe be the same throughout the whole motion. However, the velocity and direction of the motion will change because its direction is changing at every point of the circle. 

 Some examples of circular motion – 

• An artificial satellite orbiting the earth at a constant height

• a stone that is tied to a rope and is being swung in circles

• a car is turning through a curve on a race track. 

Types of Circular Motion 

 There are two types of circular motion mentioned below:

1. Uniform circular motion

2. Non-uniform circular motion.

What is a Uniform Circular Motion? 

When the motion of an object in a circle is at a constant speed, it is called uniform circular motion. An object that moves in a circle is constantly changing its direction. This means that an object that is forming a circle on a path along with that particular object will complete repeated strips around the path in the same amount of period at every time. For instance, the object is moving at a tangent to the circle. 

Some Examples of Uniform Circular Motion Are- 

1. The rotation of the blades of the ceiling fan.

2. A racing car is turning through a curve on the racing track.

3. A ball rolling on a floor with a constant velocity

4. An artificial satellite orbiting the earth at a constant height.

5. Different planets revolving at their own pace around the sun or other planets.

What is a Non-Uniform Circular Motion? 

When a motion of an object denotes changes in the speed of the particle moving along the circular path, it is called non-uniform circular motion. 

The changes that take place in the direction are accounted for by radial acceleration which is given by the given equation. 

ar = \[\frac {v^2}{r}\]

Changes that take place in the speed have implications for radial acceleration. There may be two possibilities that are as follows:

1. The radius of the circle is constant just like in the motion along a circular rail or motor track. Changes that take place in v change the magnitude of the radial acceleration. This simply means that the centripetal acceleration is nonstop. The object moving at high speed will need a greater radial which forces to change the direction and vice-versa when the radius of the circular path is constant.

2. In another possibility, the radial (centripetal) force is constant (like – a satellite rotating about the earth under the influence of a constant force of gravity). The circulation motion is fixed in its radius concerning speed changes. This simply means that unlike in the case of uniform motion, the radius of the circular path is variable. Eventually, the equation of the centripetal acceleration must be satisfied in terms of ‘speed’ and ‘radius’. However, the most important thing that is to be noted is that the change in speed of the particle affects the radial acceleration but the speed changes are not affected by the radial or centripetal force.

Some examples of non-uniform circular motion:

1. A bouncing ball.

2. A horse running in a race.

3. A bus on its way through the market.

4. Dragging a box from a path.

5. Movement of an asteroid.

6. Aircraft moving through the clouds and then landing.

In the above mention examples, the speed is varying. 

Kinematics of Circular Motion

1. Angular Position-

The angle which is made by the position vector with the given line is called angular position. Two-dimensional motion or motion in a plane is a circular motion. 

For instance- Suppose a particle P is moving in a circle having radius r and center O. The position of the object P at a given instant may be expressed by the angle q between OP and OX. This q angle is called an angular position of the particle. As the object moves in the circle its angular position q change. 

2. Angular Displacement-

An angle rotated by a position vector of the moving object in the mentioned time interval with some reference line is called angular displacement.

Important Points of Angular Displacement:

• Angular displacement is dimensionless and has proper SI unit radian while other units are degrees or revolution 2p rad =3600 1 rev.

• Finite angular displacement is not a vector quantity while infinity small angular displacement is a vector displacement because the addition of the small angular displacement is commutative while for large angular displacement it is not.

• Direction of the small angular displacement is decided by the ‘Right-Hand Thumb Rule’. When the figures are conducted along with the motion of the point then the thumb will show the direction of the angular displacement.

• It can be different for various observers.

3. Angular Velocity-

1. Average angular velocity

 w = total angle of rotation/ total time taken

Therefore; \[\frac {\theta_2 - \theta_1}{t_2 - t_1} = \frac {\Delta \theta}{\Delta t}\]

Where θ₁ & θ₂ are the angular positions of the particle at times t₁ & t₂ respectively.

2. Instantaneous angular velocity

The rate at which the position vector of a particle concerning the center rotates is called instantaneous angular velocity concerning the center. 

 w =  \[\frac {\Delta \theta}{\Delta t} = \frac {d \theta}{dt}\]

Important points:

• Angular velocity is an axial vector with dimensions (T-1) and SI unit rad/s.

• Angular velocity is known as a characteristic of the body as a whole e.g. angular velocity of all points of earth about its axis is (2p/24) rad/ hr, for a solid body as all points will rotate through the same angle in the same time.

• If a body takes ‘t’ seconds to make ‘n’ rotations then angular velocity in radian per second will be wav =2r/t

Where, t = period and f = frequency of uniform circular motion

Wav = 2π x \[\frac {1}{T}\] = 2πf 

 Where θ = a – bt + ct2 then w = \[\frac {d \theta}{dt}\] = -b + 2ct 

4. Relative Angular Velocity-

Angular velocity is described as the point from which the position vector of the moving object is drawn. The angular velocity of the particle w.r.t ‘O’ and ‘A’ will be different here. It is mentioned as below:

 WPO = \[\frac {dα}{dt}\]

 Therefore, wPA =  \[\frac {dβ}{dt}\]

5. Dynamic Circular Motion-

In two perpendicular directions, Newton’s law is applied in a circular motion. The first direction is along with the tangent and the other direction is perpendicular to it (towards the center). The component of force acting along the tangent is called the tangential (Ft) and the component of force acting at the mid is called the centripetal force (Fc). 

 Tangential force Ft = M at = M \[\frac {dv}{dt}\] = Mαr 

Centripetal force Fc = M at = mv² r = m \[\frac {v^2}{r}\]

Note: If the centripetal force is absent, the object will move in a straight line with constant speed. 

Instantaneous circle (R) has radius of:

R = \[\frac{[1+(\frac{dy}{dx})^2]^{\frac {3}{2}}}{|\frac{d^2y}{dx^2}|}\]

Key Points From The Chapter:

• Both centrifugal and centripetal forces are equal in magnitude and opposite in directions.

• These both centripetal and centrifugal forces cannot be termed as action and reaction since action and reaction never act on the same body.

• To complete the motion in a vertical circle is the basic requirement for a body under limited conditions, For example, the stress in the string must not vanish before it reaches the highest point. If it vanishes earlier, it will be devoid of the necessary centripetal force required to keep the body moving in a circle.

• Centrifugal force is known as the fictitious force which acts on a body, rotating with a uniform velocity in a circle and along with the radius away from the center.

Preparation Tips That will Help Students Solve Numerical Problems - 

• Write down all the required centripetal forces.

• Drawing a free body diagram of each component of the system will help you.

• For the rest of the components, you can check out according to the question part.

• Make it equal to the required centripetal force.

• Calculate the net radial force acting towards the center of the circular path.

• Along the radius and perpendicular to the radius, resolve the forces acting on the rotating particle.