Circular Motion Notes For JEE by Physics Wallah

Circular motion is an important Class 11 Physics topic for JEE Main and JEE Advanced. It connects vector motion, calculus, Newton’s laws of motion, friction, and rotational systems. Problems involving rotating tables, banked roads, and other circular systems require a clear understanding of velocity, acceleration, free-body diagrams, and radial force analysis.

PW supports JEE Physics preparation through concept-focused classes, live and recorded learning, practice questions, previous years’ questions (PYQs), and tests. These resources help you revise circular motion formulas, understand their applications, and practise different question patterns.

Definition Of Circular Motion

If a particle moves so that its distance from a fixed point remains constant, the motion is called circular motion.

The fixed point is the centre of the circular path, and the constant distance is its radius. A particle may move in either of two senses:

• Clockwise

• Anticlockwise

Observer Dependence Of Clockwise And Anticlockwise Motion

The description of motion as clockwise or anticlockwise depends on the observer’s position. A rotating object may appear clockwise from one side and anticlockwise from the opposite side.

Therefore, clockwise and anticlockwise descriptions are observer-dependent.

Velocity in Circular Motion

Velocity is a vector quantity with both magnitude and direction. In circular motion, the direction of velocity changes continuously as the particle moves along the circular path.

Therefore, the velocity vector cannot remain constant during circular motion, even when the speed is constant.

At every instant, the velocity acts along the tangent to the path. This principle applies to circular motion and other curved paths.

For a particle moving in a circle of radius r with angular velocity ω: v = rω

where:

• v = linear speed

• r = radius of the circular path

• ω = angular velocity

Acceleration in Circular Motion

Acceleration describes the rate of change of velocity. Since velocity continuously changes direction in circular motion, acceleration is present even when the speed remains constant.

Acceleration has two components:

Centripetal acceleration acts towards the centre of the circular path.

Uniform and Non-Uniform Circular Motion

In uniform circular motion, the centripetal acceleration is: aₙ = v²/r

Using v = rω: aₙ = rω²

The acceleration is directed towards the centre.

In non-uniform circular motion:

• Tangential acceleration changes the speed.

• Centripetal acceleration changes the direction of velocity.

• Net acceleration is the vector sum of both components.

The magnitude of tangential acceleration is: aₜ = rα

where α is angular acceleration.

When the radial and tangential components are perpendicular, the magnitude of total acceleration is: a = √(aₙ² + aₜ²)

Angular Variables

Circular motion can be described using angular quantities. As a particle moves, the radius vector joining it to the centre changes direction. Angular displacement, angular velocity, and angular acceleration describe this motion.

Angular Displacement

Angular displacement is the angle through which the radius vector turns as a particle moves from one position to another.

• Symbol: θ

• SI unit: radian (rad)

For one complete revolution: θ = 2π rad

For N complete revolutions: θ = 2πN

Angular displacement can be positive or negative depending on the chosen sign convention.

Angular Velocity

Angular velocity is the rate of change of angular displacement.

Instantaneous angular velocity: ω = dθ/dt

Average angular velocity: ω_avg = Δθ/Δt

SI unit: radian per second (rad/s).

Average angular velocity is the total angular displacement divided by the total time taken.

Angular Vectors and the Right-Hand Rule

Angular velocity and angular acceleration are axial vectors. Their directions are perpendicular to the plane of rotation.

Using the right-hand rule:

1. Curl the fingers of your right hand in the direction of rotation.

2. The extended thumb indicates the direction of the angular velocity vector.

For a chosen viewing direction:

• Anticlockwise rotation corresponds to an angular vector directed towards the observer.

• Clockwise rotation corresponds to an angular vector directed away from the observer.

These directions reverse if the viewing position changes.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity.

Instantaneous angular acceleration: α = dω/dt

Average angular acceleration: α_avg = Δω/Δt

SI unit: radian per second squared (rad/s²).

If angular velocity and angular acceleration have the same sign, angular speed increases. If they have opposite signs, angular speed decreases.

Angular Kinematics

Angular kinematics follows a mathematical structure similar to linear kinematics.

The corresponding differential relations are:

v = dx/dt

ω = dθ/dt

a = dv/dt

α = dω/dt

For circular motion with a fixed radius:

v = rω

aₜ = rα

Equations for Constant Angular Acceleration

When angular acceleration is constant, the following equations apply:

ω = ω₀ + αt

θ = θ₀ + ω₀t + ½αt²

ω² = ω₀² + 2α(θ − θ₀)

where:

• ω₀ = initial angular velocity

• ω = angular velocity after time t

• α = constant angular acceleration

• θ₀ = initial angular position

• θ = final angular position

These equations are valid only when angular acceleration remains constant.

Angular Velocity, Frequency, and RPM

One complete revolution equals 2π radians. If N is the number of revolutions: θ = 2πN

Angular velocity and frequency are related by: ω = 2πf

where f is the frequency in revolutions per second.

If rotational speed is given in revolutions per minute (RPM): ω = 2π × RPM/60

Memory tip: Multiply revolutions per second by 2π directly. For RPM, divide by 60 first.

Variable Angular Acceleration

When angular acceleration varies with time, use integration:

dω = α dt

dθ = ω dt

Therefore, ω(t) = ω₀ + ∫₀ᵗ α(t) dt

For angular position: θ(t) = θ₀ + ∫₀ᵗ ω(t) dt

For example, if α = 4t² + 2t and the body starts from rest, then:

ω(2) = ∫₀² (4t² + 2t) dt

ω(2) = [4t³/3 + t²]₀²

ω(2) = 32/3 + 4 = 44/3 rad/s

Relative Angular Velocity

The angular velocity of a particle as observed from another point depends on the component of relative velocity perpendicular to the line joining the observer and the particle.

For a particle at position vector r relative to an observer, its angular velocity vector is:

ω⃗ = (r⃗ × v⃗)/r²

The magnitude is: ω = v⊥/r

where v⊥ is the component of velocity perpendicular to the position vector.

For two particles A and B, the relative angular velocity of A about B is determined by the relative position and relative velocity:

ω⃗₍A/B₎ = (r⃗₍A/B₎ × v⃗₍A/B₎)/|r⃗₍A/B₎|²

Here,

r⃗₍A/B₎ = r⃗ₐ − r⃗ᵦ

v⃗₍A/B₎ = v⃗ₐ − v⃗ᵦ

The parallel component of relative velocity changes the separation between the particles, while the perpendicular component changes the viewing angle.

For a particle moving in a circle of radius R with speed v:

ω = v/R

For the same velocity magnitude, if the perpendicular distance from an observation point is 2R, the angular speed magnitude is v/(2R), provided the velocity is perpendicular to the position vector.

Similarly, if the perpendicular distance is 3R under the same condition, the angular speed magnitude is v/(3R).

Dynamics of Circular Motion

Circular motion dynamics involves applying Newton’s second law to the forces acting on a particle.

Follow these steps to solve a problem:

1. Identify the plane of motion.

2. Mark the centre and radius of the circular path.

3. Draw a free-body diagram showing all actual forces.

4. Resolve the forces into radial and tangential components.

5. Apply Newton’s second law separately along each direction.

The net inward radial force must provide centripetal acceleration: ΣFᵣ = mv²/r

Using v = rω: ΣFᵣ = mrω²

The net tangential force produces tangential acceleration: ΣFₜ = maₜ

Since aₜ = rα: ΣFₜ = mrα

Centripetal force is not a separate type of force. It is the net inward force provided by actual forces such as tension, friction, normal reaction, gravity, spring force, or their combination.

For motion constrained to a plane, the net force perpendicular to that plane must be zero when there is no acceleration in that direction.

Rotating Table and Static Friction

Consider a block placed at a distance l from the centre of a horizontal table rotating with constant angular speed ω.

The required centripetal force is: Fᶜ = mω²l

For a horizontal table, assuming no vertical acceleration: N = mg

Static friction provides the required centripetal force: fₛ = mω²l

Static friction is self-adjusting. It does not always equal μN; it reaches the limiting value only when the block is about to slip.

The maximum static friction is: fₛ,max = μₛN = μₛmg

For the block not to slip: mω²l ≤ μₛmg

Therefore, ω² ≤ μₛg/l

The maximum angular speed is: ω_max = √(μₛg/l)

Rotating Table with Angular Acceleration

If the table has both angular velocity ω and angular acceleration α, friction must provide radial and tangential acceleration.

Radial acceleration: aᵣ = ω²l

Tangential acceleration: aₜ = αl

Since these components are perpendicular, the required frictional force is: f = m√[(ω²l)² + (αl)²]

Equivalently, f = ml√(ω⁴ + α²)

Slipping begins when the required friction reaches the limiting value: f = μₛmg

Banked Roads

Banking means raising the outer edge of a road on a curved path. It allows the horizontal component of the normal reaction to help provide the centripetal force.

Level Road

On a level road, friction provides the centripetal force. f = mv²/r

The maximum available friction is: f_max = μₛmg

At the limiting speed: mv_max²/r = μₛmg

Therefore, v_max = √(μₛrg)

This is the maximum speed for the vehicle to negotiate the curve without skidding, under the idealised level-road model.

Frictionless Banked Road

For a road banked at angle θ, the normal reaction N has vertical and horizontal components.

Vertical balance: N cos θ = mg

Horizontal component providing centripetal force: N sin θ = mv²/r

Dividing the equations gives: tan θ = v²/(rg)

Therefore, the optimum speed is: v₀ = √(rg tan θ)

At this speed, friction is not required in the idealised model.

Banked Road with Friction

When friction is present, the permissible speed depends on the direction of friction.

If the vehicle moves faster than the frictionless optimum speed, it tends to move up the bank. Friction acts down the slope.

The maximum speed is: v_max² = rg(tan θ + μₛ)/(1 − μₛ tan θ)

Thus, v_max = √[rg(tan θ + μₛ)/(1 − μₛ tan θ)]

If the vehicle moves slower than the optimum speed, it tends to slide down the bank. Friction acts up the slope.

The minimum speed is: v_min² = rg(tan θ − μₛ)/(1 + μₛ tan θ)

Thus, v_min = √[rg(tan θ − μₛ)/(1 + μₛ tan θ)]

These expressions assume the standard banked-road model with static friction at its limiting value. The minimum-speed expression applies when tan θ ≥ μₛ; otherwise, the idealised model permits the vehicle to remain at rest without sliding down.

Memory tip: For maximum speed, use plus in the numerator and minus in the denominator. For minimum speed, use minus in the numerator and plus in the denominator.

Death Well

A death well is a vertical cylindrical structure in which a vehicle moves along the inner wall in a horizontal circle. The wall’s normal reaction provides centripetal force, while friction prevents the vehicle from sliding downward.

For a vehicle moving at speed v in a circular path of radius r: N = mv²/r

The maximum upward friction is: fₛ,max = μₛN

For the vehicle to avoid sliding down: μₛN ≥ mg

Substituting N = mv²/r: μₛmv²/r ≥ mg

After cancelling m: v² ≥ rg/μₛ

Therefore, the minimum speed is: v_min = √(rg/μₛ)

This result assumes the friction coefficient is μₛ and the vehicle is in contact with the vertical wall.

Radius of Curvature

The radius of curvature describes how tightly a path bends at a particular point. A smaller radius indicates a sharper curve.

For motion along a smooth curved path, the normal component of acceleration is: aₙ = v²/r

Therefore, r = v²/aₙ

where:

• r = radius of curvature

• v = instantaneous speed

• aₙ = magnitude of the acceleration component perpendicular to the velocity

For a projectile moving under gravity alone, the radius of curvature is smallest at the highest point of its trajectory. At that point, the vertical component of velocity is zero, so the speed is at its minimum, while gravity acts perpendicular to the horizontal velocity.

If the projectile’s horizontal velocity is uₓ, then at the highest point: v = |uₓ|

Since the normal acceleration equals g there, r = uₓ²/g

Circular motion connects linear velocity, angular variables, acceleration, and Newton’s laws. For JEE Main and JEE Advanced, focus on the relationships v = rω, aₙ = v²/r, and aₜ = rα, along with radial force analysis and friction-based applications. Practise drawing free-body diagrams before substituting values into formulas. PW supports your JEE Physics preparation with concept-focused classes, live and recorded learning, practice questions, PYQs, and tests to help you revise formulas and apply them to different problem types.

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