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Circular Motion | Physics Class 11 - NEET

Circular Motion

  • When a particle moves in a plane and keeps the same distance from a fixed or moving point, this type of movement is called circular motion.
  • The fixed or moving point that the particle is measured from is known as the center.
  • The distance between the fixed point and the particle is referred to as the radius.

For Example: The car is moving in a straight line with respect to the man A. But the man B continuously rotate his face to see the car. So with respect to man A Circular Motion | Physics Class 11 - NEET. But with respect to man B Circular Motion | Physics Class 11 - NEET. Therefore we conclude that with respect to A the motion of car is straight line but for man B it has some angular velocity.

Circular Motion | Physics Class 11 - NEET

Analogy between Linear Kinematics & Circular Kinematics

Circular Motion | Physics Class 11 - NEET

Question for Circular Motion
Try yourself:
Which of the following best describes circular motion?
View Solution

Kinematics of Circular Motion

1. Angular Position 

The angle formed by the position vector with a specific reference line is known as angular position, which can be measured in either radians or degrees.

Circular Motion | Physics Class 11 - NEET

 Circular motion occurs in two dimensions, or in a plane. Imagine a particle P moving in a circle with a radius r and centre O. The position of the particle P at any moment can be described by the angle θ between the line OP and the line OX. This angle θ represents the angular position of the particle. As the particle moves around the circle, its angular position θ changes. If the particle rotates through an angle Δθ in a time Δt, this describes its motion.

2. Angular Displacement 

Angle rotated by a position vector of the moving particle in a given time interval with some reference line is called its angular displacement.

Important Points:

  • It is a vector quantity with the SI unit of radian; other units include degrees and revolutions (2π rad = 360° = 1 rev).
  • Both small and finite angular displacements are vector quantities.
  • The direction of angular displacement follows the right-hand thumb rule: if your fingers point in the direction of motion, your thumb indicates the direction of angular displacement.
  • Angular displacement can vary for different observers.
  • It relates to angular velocity, which indicates how fast an object rotates.
  • Angular displacement is essential for understanding the rotational motion of an object around a fixed axis.

3. Angular velocity ω

Angular velocity is the vector measure of the rotation rate, which refers to how fast an object rotates or revolves relative to another point.

Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

where θ1 and θ2 are angular position of the particle at time t1 and t2 respectively.

 Instantaneous Angular Velocity

The rate at which the position vector of a particle with respect to the centre rotates, is called as instantaneous angular velocity with respect to the centre.

Circular Motion | Physics Class 11 - NEET

Important points:

  • It is an axial vector with dimensions [T-1] and SI unit rad/s.
  • For a rigid body as all points will rotate through same angle in same time, angular velocity is a characteristic of the body as a whole, e.g., angular velocity of all points of earth about its own axis is (2π/24) rad/hr.
  • If a body makes `n' rotations in `t' seconds then angular velocity in radian per second will beCircular Motion | Physics Class 11 - NEET
    If T is the period and ‘f’ the frequency of uniform circular motion.
    Circular Motion | Physics Class 11 - NEET
  • If θ = a – bt + ct2 then ω = dθ/dt = – b + 2ct

Relation Between Speed and Angular Velocity

Circular Motion | Physics Class 11 - NEET

The rate of change of angular velocity is called the angular acceleration (α). Thus,

Circular Motion | Physics Class 11 - NEET

The linear distance PP' travelled by the particle in time Δt is

Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

Here, v is the linear speeds of the particle. It is only valid for circular motion.

v = rω is a scalar quantity.

Question for Circular Motion
Try yourself:
Which quantity represents the rate at which an object rotates or revolves relative to another point?
View Solution

4. Relative Angular Velocity

Angular velocity is defined with respect to the point from which the position vector of the moving particle is drawn. Here angular velocity of the particle w.r.t. 'O' and 'A' will be different.

Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

Relative angular velocity of a particle ‘A’ with respect to the other moving particle ‘B’ is the angular velocity of the position vector of ‘A’ with respect to ‘B’. That means it is the rate at which position vector of ‘A’ with respect to ‘B’ rotates at that instant

Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

Important Points:

  • If two particles are moving on the same circle or different coplanar concentric circles in same direction with different uniform angular speed ωA and ωB respectively, the rate of change of angle between OA and OB is
    Circular Motion | Physics Class 11 - NEET
    Circular Motion | Physics Class 11 - NEET
    So the time taken by one to complete one revolution around O w.r.t., the other
    Circular Motion | Physics Class 11 - NEET
  • If two particles are moving on two different concentric circles with different velocities then angular velocity of B relative to A as observed by A will depend on their positions and velocities. Consider the case when A and B are closest to each other moving in same direction as shown in figure. In this situation
    Circular Motion | Physics Class 11 - NEET
    Circular Motion | Physics Class 11 - NEET
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5. Angular Acceleration α

  •  Average Angular Acceleration

Let ω1 and ω2 be the instantaneous angular speeds at times t1 and t2 respectively, then the average angular acceleration αav is defined as

Circular Motion | Physics Class 11 - NEET

  •  Instantaneous Angular Acceleration

It is the limit of average angular acceleration as ∆t approaches zero, i.e.,

Circular Motion | Physics Class 11 - NEET

Important points:

  • It is also an axial vector with dimension [T-2] and unit rad/s2
  • If α = 0, circular motion is said to be uniform.
  •  AsCircular Motion | Physics Class 11 - NEET
    i.e., second derivative of angular displacement w.r.t time gives angular acceleration.
  • α is an axial vector, and its direction is the same as ω if ω increases and opposite to ω if ω decreases.
  • The direction of angular acceleration is along the axis of rotation.

Types of Circular Motion

For better understanding, we can classify the circular motion in following three types:

1. Uniform circular motion in which v and ω are constant.

In this motion,Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

a is towards centre, v is tangential and according to the shown figure, ω is perpendicular to paper inwards or in ⊗ direction.

2. Accelerated circular motion in which v and ω are increasing. So, ais in the direction of v and α is in the direction of ω.

In the figure shown, α and ω both are perpendicular to paper inwards. Further,

Circular Motion | Physics Class 11 - NEET

and Circular Motion | Physics Class 11 - NEET

3. Retarded circular motion in which v and ω are decreasing. So, at is in the opposite direction of v and α is in the opposite direction of ω.

In the figure shown, ω is perpendicular to paper inwards in ⊗direction and α is perpendicular to paper outwards in O direction.

Circular Motion | Physics Class 11 - NEET

Note: In the above figures, θ is the angle between v and a.

Dynamics of Circular Motion


We know that linear acceleration of a particle in circular motion has two components, tangential and radial (or centripetal). So, normally we resolve the forces acting on the particle in two directions:

(i) Tangential

(ii) Radial

Circular MotionCircular Motion

In tangential direction, net force on the particle is maand in radial direction net force is mar. In uniform circular motion, tangential acceleration is zero. Hence, net force in tangential direction is zero and in radial direction

Circular Motion | Physics Class 11 - NEET

This net force (towards centre) is also called centripetal force.

In most of the cases plane of our uniform circular motion will be horizontal and one of the tangent is in vertical direction also. So, in this case we resolve the forces in:

  •  horizontal radial direction
  •  vertical tangential direction

In vertical tangential direction net force is zero (at =0) and in horizontal radial direction (towards centre) net force is mv2/r or mrω2

Note:  Centripetal force mv2/r or mrω2 (towards centre) does not act on the particle but this much force is required to the particle for rotating in a circle (as it is accelerated due to change in direction of velocity). The real forces acting on the particle provide this centripetal force or we can say that vector sum of all the forces acting on the particle is equal to mv2/r  or mrω2 (in case of uniform circular motion). The real forces acting on the particle may be, friction force, weight, normal reaction, tension etc.

Question for Circular Motion
Try yourself:
What is the direction of angular acceleration if the angular speed is decreasing in a clockwise circular motion?
View Solution

Conical Pendulum

If a small particle of mass m tied to a string is whirled in a horizontal circle, as shown in Fig. The arrangement is called the ‘conical pendulum’. In case of conical pendulum, the vertical component of tension balances the weight in tangential direction, while its horizontal component provides the necessary centripetal force in radial direction

(towards centre). Thus,

Circular Motion in a PendulumCircular Motion in a Pendulum

Circular Motion | Physics Class 11 - NEET

From these two equations, we can find

∴ Angular speed   Circular Motion | Physics Class 11 - NEET

So, the time period of pendulum is

Circular Motion | Physics Class 11 - NEET         (as r = L cos θ)

Motion of a Particle Inside a Smooth Cone

A particle of mass ‘m’ is rotating inside a smooth cone in horizontal circle of radius ‘r’ as shown in figure. constant speed of the particle is suppose ‘v’. Only two forces are acting on the particle in the shown directions:

(i) Normal reaction N

(ii) Weight mg

We have resolved these two forces in vertical tangential direction and horizontal radial direction. In vertical tangential direction, net force is zero.

Motion of Particle Inside a ConeMotion of Particle Inside a Cone

∴ N cosθ = mg

In horizontal radial direction (towards centre), net force is mv2/r

∴ Circular Motion | Physics Class 11 - NEET

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Circular Motion
Download as PDF
Download as PDF

Circular Turning on Roads

When vehicles go through turnings, they travel along a nearly circular arc. There must be some force which will produce the required centripetal acceleration. If the vehicles travel in a horizontal circular path, this resultant force is also horizontal. 

The necessary centripetal force is being provided to the vehicles by following three ways:

1. By Friction only

2. By Banking of Roads only

3. By Friction and Banking of Roads both.

In real life the necessary centripetal force is provided by friction and banking of roads both. Now let us write equations of motion in each of the three cases separately and see what are the constant in each case.

  •  By Friction Only 

Suppose a car of mass m is moving at a speed v in a horizontal circular arc of radius r. In this case, the necessary centripetal force to the car will be provided by force of friction f acting towards centre

Thus, f = mv2/r

Further, limiting value of f is or μN or fL = μN = μmg (N = mg)

Therefore, for a safe turn without sliding  mv2/r ≤ fL

or Circular Motion | Physics Class 11 - NEET

Here, two situations may arise. If m and r are known to us, the speed of the vehicle should not exceed Circular Motion | Physics Class 11 - NEET and if v and r are known to us, the coefficient of friction should be greater than Circular Motion | Physics Class 11 - NEET.

  •  By Banking of Roads Only 

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

Circular Motion | Physics Class 11 - NEET

Applying Newton's second law along the radius and the first law in the vertical direction.

Circular Motion | Physics Class 11 - NEET

from these two equations, we get

Circular Motion | Physics Class 11 - NEET

  • By Friction and Banking of Road Both

If a vehicle is moving on a circular road which is rough and banked also, then three forces may act on the vehicle, of these force, the first force, i.e., weight (mg) is fixed both in magnitude and direction.

Circular Motion | Physics Class 11 - NEET

The direction of second force i.e., normal reaction N is also fixed (perpendicular or road) while the direction of the third i.e., friction f can be either inwards or outwards while its magnitude can be varied upto a maximum limit (fL = mN). So the magnitude of normal reaction N and directions plus magnitude of friction f are so adjusted that the resultant of the three forces mentioned above is mv2/r towards the center. Of these m and r are also constant. Therefore, magnitude of N and directions plus magnitude of friction mainly depends on the speed of the vehicle v. Thus, situation varies from problem to problem. Even though we can see that :

(i) Friction f will be outwards if the vehicle is at rest v = 0. Because in that case the component weight mg sinθ is balanced by f.

(ii) Friction f will be inwards if

v > Circular Motion | Physics Class 11 - NEET

(iii) Friction f will be outwards if

v < Circular Motion | Physics Class 11 - NEET and

(iv) Friction f will be zero if

v = Circular Motion | Physics Class 11 - NEET

(v) For maximum safe speed 

Circular Motion | Physics Class 11 - NEET

As maximum value of friction f = μN
Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

Similarly; Circular Motion | Physics Class 11 - NEET

  • The expression tanθ = Circular Motion | Physics Class 11 - NEET also gives the angle of banking for an aircraft, i.e., the angle through which it should tilt while negotiating a curve, to avoid deviation from the circular path.  
  • The expression tanθ = Circular Motion | Physics Class 11 - NEET also gives the angle at which a cyclist should lean inward when rounding a corner. In this case, θ is the angle which the cyclist must make with the vertical to negotiate a safe turn.

Question for Circular Motion
Try yourself:A car of mass 1000 kg is moving on a circular road of radius 50 meters. If the coefficient of friction between the tires and the road is 0.4, what is the maximum speed the car can have without sliding?
View Solution

Death well

A motor cyclist is driving in a horizontal circle on the inner surface of vertical cylinder of radius R. Friction coefficient between tyres of motorcyclist and surface of cylinder is m. Find out the minimum velocity for which the motorcyclist can do this. v is the speed of motor cyclist and m is his mass.

Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

Cyclist does not drop down when

Circular Motion | Physics Class 11 - NEET

Motion of a Cyclist on a Circular Path

Suppose a cyclist is going at a speed v on a circular horizontal road of radius r which is not banked. Consider the cycle and the rider together as the system. The centre of mass C (figure shown) of the system is going in a circle with the centre at O and radius r.

Circular Motion | Physics Class 11 - NEET

Let us choose O as the origin, OC as the X-axis and vertically upward as the Z-axis. This frame is rotating at an angular speed w = v/r about the Z-axis. In this frame the system is at rest. Since we are working from a rotating frame of reference, we will have to apply a centrifugal force on each particle. The net centrifugal force on particle will be Mω2r = Mv2/r, where M is the total mass of the system. This force will act through the centre of mass. Since the system is at rest in this frame, no other pseudo force is needed.

Figure in shows the forces. The cycle is bent at an angle θ with the vertical. The forces are

(i) weight Mg,

(ii) normal force N

(iii) friction f

(iv) Centrifugal force 

In the frame considered, the system is at rest. Thus, the total external force and the total external torque must be zero. Let us consider the torques of all the forces about the point A. The torques of N and f about A are zero because these forces pass through A. The torque of Mg about A is Mg(AD) in the clockwise direction and that of Mv2/r is Mv2/r (CD) in the anticlockwise direction. For rotational equilibrium,

Mg(AD) = Mv2/r (CD)

Circular Motion | Physics Class 11 - NEET

Circular Motion | Physics Class 11 - NEET

Thus, the cyclist bends at an angleCircular Motion | Physics Class 11 - NEETwith the vertical.

Centrifugal Force

When a body is rotating in a circular path and the centripetal force vanishes, the body would leave the circular path. To an observer A who is not sharing the motion along the circular path, the body appears to fly off tangentially at the point of release. To another observer B, who is sharing the motion along the circular path (i.e., the observer B is also rotating with the body which is released, it appears to B, as if it has been thrown off along the radius away from the centre by some force. This inertial force is called centrifugal force.)

Its magnitude is equal to that of the centripetal force = mv2/r . Centrifugal force is a fictitious force which has to be applied as a concept only in a rotating frame of reference to apply Newton's law of motion in that frame)

FBD of ball w.r.t non-inertial frame rotating with the ball.

Circular Motion | Physics Class 11 - NEET

Suppose we are working from a frame of reference that is rotating at a constant, angular velocity ω with respect to an inertial frame. If we analyse the dynamics of a particle of mass m kept at a distance r from the axis of rotation, we have to assume that a force mrω2 act radially outward on the particle. Only then we can apply Newton's laws of motion in the rotating frame. This radially outward pseudo force is called the centrifugal force.

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Simple Pendulum 

A simple pendulum is constructed by attaching a bob of mass m to a string of length L fixed at its upper end. The bob oscillates in a vertical circle. It is found that the speed of the bob is v when the string makes an angle θ with the vertical.

The force acting on the bob are (figure)

Circular Motion | Physics Class 11 - NEET

(i) Tension T

(ii) Weight mg

As the bob moves in a vertical circle with centre at O, the radial acceleration is v2/L towards O. Taking the components along this radius and applying Newton's second law, we get

Circular Motion | Physics Class 11 - NEET

Circular Motion in Horizontal Plane 

A ball of mass m attached to a light and inextensible string rotates in a horizontal circle of radius r with an angular speed w about the vertical. If we draw the force diagram of the ball. 

Circular Motion | Physics Class 11 - NEET

We can easily see that the component of tension force along the centre gives the centripetal force and component of tension along vertical balances the gravitation force. Such a system is called a conical pendulum.

We already know a bit about circular motion. It's a kind of movement where an object stays at the same distance from a fixed point. Circular motion is divided into two types: uniform and non-uniform. Examples include swinging a ball on a string or a car turning on a track. Now, let's talk about a special kind of circular motion called vertical circular motion.

Vertical Circular Motion

Imagine an object connected to a string and rotated around in a circle vertically.

Vertical Circular MotionVertical Circular Motion

Between points X and Y, the tension in the string balances the weight, keeping it taut. To find the velocity needed to reach Y, we use conservation of mechanical energy. 

Circular Motion | Physics Class 11 - NEET

Since the particle just reaches Y, its velocity at Y is zero. 

Circular Motion | Physics Class 11 - NEET

By equating the initial and final energy, we determine the velocity.

Circular Motion | Physics Class 11 - NEET

Now, if we want to find the minimum velocity to reach Z, assuming zero velocity at Z is incorrect. Velocity at Z should be such that weight equals centripetal force, making tension zero.

Circular Motion | Physics Class 11 - NEET

Substituting the velocity value, 

Circular Motion | Physics Class 11 - NEET

Now we equate energies for X and Z. 

Circular Motion | Physics Class 11 - NEET

The critical values lead to three cases:

  • The ball oscillates and never reaches Y.

Circular Motion | Physics Class 11 - NEET

  • The ball loses contact between Y and Z, starting projectile motion.

Circular Motion | Physics Class 11 - NEET

  •  The string never becomes slack, completing the circle.

Circular Motion | Physics Class 11 - NEET

This situation is similar for a ball tied to a rod moving in a vertical circle. The only difference is that the velocity at the top can be zero, as the normal reaction of the rod balances the weight. 

Solving similarly yields three cases for a rod:

  • The body oscillates and doesn't reach Y.

Circular Motion | Physics Class 11 - NEET

  • The ball oscillates, crosses Y, but doesn't reach Z.

Circular Motion | Physics Class 11 - NEET

  • The body completes the circle.

Circular Motion | Physics Class 11 - NEET

Solved Examples on Vertical Circular Motion

Q1. A particle travels in a circle of radius 20 cm at a speed that uniformly increases. If the speed changes from 5·0 m/s to 6·0 m/s in 2·0 s, find the angular acceleration.

Sol: The tangential acceleration is given by

Circular Motion | Physics Class 11 - NEET

Q2. A small block of mass 100 g moves with uniform speed in a horizontal circular groove, with vertical side walls, of radius 25 cm. If the block takes 2·0 s to complete one round, find the normal contact force by the side wall of the groove.

Sol: The speed of the block is

Circular Motion | Physics Class 11 - NEET

The acceleration of the block is

Circular Motion | Physics Class 11 - NEET

towards the centre. The only force in this direction is
the normal contact force due to the side walls. Thus,
from Newton’s second law, this force is

F=ma=(0⋅100 kg) (2⋅5 m/s2)=0⋅25 N.

Q3. A particle of mass m is suspended from a ceiling through a string of length L. The particle moves in a horizontal circle of radius r. Find
 (a) the speed of the particle and
 (b) the tension in the string. Such a system is called a conical pendulum.Circular Motion | Physics Class 11 - NEET

Sol: The situation is shown in figure (7-W2). The angle θ made by the string with the vertical is given by

sin⁡θ = r/L.                                                           .....(i)

The forces on the particle are
(a) the tension T along the string and
(b) the weight mg vertically downward.

The particle is moving in a circle with a constant speed v. Thus, the radial acceleration towards the centre has magnitude v2/r. Resolving the forces along the radial direction and applying Newton’s second law,

T sin⁡θ = m(v2/r).                                                       ....(ii)

As there is no acceleration in vertical direction, we have from Newton’s first law,

Tcosθ=mg.

Circular Motion | Physics Class 11 - NEETT \sin\theta = m \left(\frac{v^2}{r}\right).

\sin\theta = \frac{r}{L}.

The document Circular Motion | Physics Class 11 - NEET is a part of the NEET Course Physics Class 11.
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FAQs on Circular Motion - Physics Class 11 - NEET

1. What is the difference between linear kinematics and circular kinematics?
Ans.Linear kinematics deals with the motion of objects along a straight path, characterized by parameters such as distance, displacement, speed, velocity, and acceleration. In contrast, circular kinematics focuses on the motion of objects along a circular path, where parameters include angular displacement, angular velocity, and angular acceleration. While both types of motion involve similar concepts, circular motion introduces additional factors due to the curvature of the path.
2. What are the types of circular motion?
Ans.The types of circular motion include uniform circular motion and non-uniform circular motion. In uniform circular motion, an object travels around a circular path at a constant speed, meaning that its angular velocity remains constant. In non-uniform circular motion, the object travels along a circular path while its speed or direction changes, resulting in varying angular velocity.
3. How does a conical pendulum work?
Ans.A conical pendulum consists of a mass attached to a string that swings in a horizontal circle while moving vertically. The tension in the string provides the necessary centripetal force to keep the mass moving in a circular path, while the weight of the mass acts vertically downward. The motion creates a conical shape as the pendulum swings, with the angle between the string and the vertical axis determining the radius of the circular path.
4. What is the effect of circular turning on roads for vehicles?
Ans.Circular turning on roads requires vehicles to exert a centripetal force to maintain a curved path. The friction between the tires and the road surface provides this force. If the speed of the vehicle is too high or the road is too slippery, the friction may not be sufficient, resulting in skidding or losing control. Engineers design road curves with banking to help vehicles negotiate turns safely by reducing the reliance on friction.
5. What happens in a death well in terms of circular motion?
Ans.A death well is a vertical cylindrical structure where participants slide down and are able to loop around the inner surface without falling. This phenomenon occurs due to centripetal force acting on the participants as they descend, which allows them to maintain circular motion despite gravitational forces. The key is achieving the necessary speed at the bottom of the well to ensure that the centripetal force equals or exceeds the gravitational force acting on the participants.
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