Uniform Circular Motion Formula Notes for NEET Students

 Uniform Circular Motion Formula Notes for NEET Students

-  Dr.Sanjaykumar pawar

UNIFORM CIRCULAR MOTION

│

├── Centripetal Acceleration

│   │

│   ├── Definition

│   │   └── Acceleration towards centre

│   │

│   ├── Formula

│   │   └── ac = v² / R

│   │

│   ├── Direction

│   │   ├── Towards centre

│   │   └── Centre-seeking

│   │

│   ├── Nature

│   │   ├── Magnitude constant

│   │   ├── Direction changes

│   │   └── Not a constant vector

│   │

│   └── Proposed By

│       ├── Isaac Newton

│       └── Christiaan Huygens

│

├── Angular Distance

│   │

│   ├── Symbol

│   │   └── Δθ

│   │

│   └── Meaning

│       └── Angle turned by radius vector

│

├── Angular Speed (ω)

│   │

│   ├── Definition

│   │   └── Rate of change of angular displacement

│   │

│   ├── Formula

│   │   └── ω = Δθ / Δt

│   │

│   ├── SI Unit

│   │   └── rad s⁻¹

│   │

│   └── Relation with Frequency

│       └── ω = 2πν

│

├── Linear Speed (v)

│   │

│   ├── Formula

│   │   └── v = Δs / Δt

│   │

│   ├── Arc Length Relation

│   │   └── Δs = RΔθ

│   │

│   └── Relation with Angular Speed

│       └── v = Rω

│

├── Centripetal Acceleration Using Angular Speed

│   │

│   └── Formula

│       └── ac = ω²R

│

├── Time Period (T)

│   │

│   ├── Definition

│   │   └── Time for one revolution

│   │

│   └── Relation

│       └── T = 1/ν

│

├── Frequency (ν)

│   │

│   ├── Definition

│   │   └── Revolutions per second

│   │

│   ├── Formula

│   │   └── ν = 1/T

│   │

│   └── Unit

│       └── Hertz (Hz)

│

├── One Complete Revolution

│   │

│   ├── Distance Covered

│   │   └── s = 2πR

│   │

│   ├── Speed Formula

│   │   └── v = 2πR / T

│   │

│   └── Speed Using Frequency

│       └── v = 2πRν

│

├── Centripetal Acceleration Using Frequency

│   │

│   └── Formula

│       └── ac = 4π²ν²R

│

├── Important NEET Concepts

│   │

│   ├── Speed constant

│   ├── Velocity changes

│   ├── Velocity tangent to path

│   ├── Acceleration towards centre

│   ├── Angular speed measures angular motion

│   └── Circular motion is accelerated motion

│

├── Important Formula Summary

│   │

│   ├── ac = v²/R

│   ├── v = Rω

│   ├── ac = ω²R

│   ├── ω = 2πν

│   ├── ν = 1/T

│   ├── v = 2πR/T

│   ├── v = 2πRν

│   └── ac = 4π²ν²R

│

└── Memory Tricks

    │

    ├── Velocity → Tangent

    ├── Acceleration → Centre

    ├── v = Rω

    └── ac = ω²R

Uniform Circular Motion showing velocity tangent to the circle and centripetal acceleration directed towards the centre.

Internal Links

Motion in a Plane Notes

Projectile Motion Complete Notes

Laws of Motion NEET Notes

Kinematics Formula Sheet

Vectors in Physics Explained

Circular Motion MCQs for NEET

Angular Velocity and Rotational Motion

Newton’s Laws of Motion Notes

Relative Velocity Notes

Physics Chapter Wise NEET Questions

Uniform Circular Motion — Angular Speed, Time Period and Frequency

Centripetal Acceleration

When an object moves in a circular path with speed v and radius R, the acceleration acting on it is called centripetal acceleration.

a_c = v² / R

Where:

• a_c = centripetal acceleration
• v = speed of object
• R = radius of circular path

Direction of Centripetal Acceleration

• The acceleration always acts towards the centre of the circle.
• Therefore it is called centripetal acceleration.
• The word centripetal means "centre-seeking".

Although the magnitude of centripetal acceleration remains constant, its direction changes continuously. Therefore, centripetal acceleration is not a constant vector.

Why Direction Changes?

As the object moves around the circle:

• Direction of velocity changes continuously.
• Therefore acceleration direction also changes continuously.
• At every point, acceleration points towards the centre.

Historical Information

• The term centripetal acceleration was proposed by Isaac Newton.
• Detailed analysis was first published by Dutch scientist Christiaan Huygens in 1673.

Angular Distance

When an object moves from one point to another in a circular path, the radius vector turns through some angle.

This angle is called angular displacement or angular distance.

Δθ

Angular Speed

Angular speed tells us how fast the angle changes with time.

ω = Δθ / Δt

Where:

• ω = angular speed
• Δθ = angular displacement
• Δt = time interval

SI Unit of Angular Speed

radian/second (rad s^-1)

Relation Between Linear Speed and Angular Speed

Suppose:

• Distance travelled along circle = Δs

We know:

v = Δs / Δt

Also:

Δs = RΔθ

Substituting:

v = RΔθ / Δt

Therefore:

v = Rω

Centripetal Acceleration in Terms of Angular Speed

We know:

a_c = v² / R

Using:

v = Rω

Substituting:

a_c = (Rω)² / R

a_c = R²ω² / R

a_c = ω²R

Time Period (T)

The time taken by an object to complete one revolution is called time period.

T

Frequency (ν)

The number of revolutions completed in one second is called frequency.

ν

Relation Between Frequency and Time Period

ν = 1 / T

T = 1 / ν

Distance Covered in One Revolution

In one complete revolution, the object covers the circumference of the circle.

s = 2πR

Speed in Terms of Time Period

We know:

v = distance / time

Distance in one revolution:

2πR

Time taken:

T

Therefore:

v = 2πR / T

Speed in Terms of Frequency

Since:

ν = 1 / T

Therefore:

v = 2πRν

Angular Speed in Terms of Frequency

In one revolution:

Angular displacement = 2π

Therefore:

ω = 2πν

Centripetal Acceleration in Terms of Frequency

Using:

a_c = ω²R

and

ω = 2πν

Substituting:

a_c = (2πν)²R

a_c = 4π²ν²R

Important Formula Summary

Formula Name	Formula
Linear speed	v = Rω
Angular speed	ω = Δθ / Δt
Centripetal acceleration	ac = v² / R
Acceleration using angular speed	ac = ω²R
Frequency relation	ν = 1 / T
Angular speed using frequency	ω = 2πν
Linear speed using frequency	v = 2πRν
Acceleration using frequency	ac = 4π²ν²R

NEET Important Points

• Velocity acts along tangent to circle.
• Centripetal acceleration acts towards centre.
• Speed remains constant but velocity changes.
• Angular speed measures rate of angular displacement.
• Frequency means revolutions per second.
• Time period means time for one revolution.

Memory Tricks

Velocity → Tangent

Acceleration → Centre

Formula Memory Tricks

• v = Rω → "Velocity equals omega R"
• a_c = ω²R → "Acceleration equals omega square R"
• ν = 1/T → "Frequency is inverse of time period"

Angular Speed, Time Period and Frequency

Questions and Answers for CBSE Class 11

Multiple Choice Questions (MCQs)

1. The acceleration directed towards the centre of a circle is called:

A. Linear acceleration
B. Tangential acceleration
C. Centripetal acceleration
D. Gravitational acceleration

Answer: C. Centripetal acceleration

2. The formula of centripetal acceleration is:

A. vR
B. v²/R
C. R²/v
D. R/v²

Answer: B. v²/R

3. Angular speed is represented by:

A. α
B. β
C. ω
D. ν

Answer: C. ω

4. SI unit of angular speed is:

A. m/s
B. rad/s
C. m
D. Hz

Answer: B. rad/s

5. Relation between linear speed and angular speed is:

A. v = ω/R
B. v = Rω
C. v = R/ω
D. v = ω²R

Answer: B. v = Rω

Very Short Answer Questions

1. Define angular speed.

Angular speed is the rate of change of angular displacement with time.

2. Write the formula of angular speed.

ω = Δθ / Δt

3. Define time period.

The time taken to complete one revolution is called time period.

4. Define frequency.

The number of revolutions completed in one second is called frequency.

5. Write relation between frequency and time period.

ν = 1 / T

6. Write relation between linear speed and angular speed.

v = Rω

7. What is the SI unit of frequency?

Hertz (Hz)

Short Answer Questions

1. Explain angular speed.

Angular speed is the angle covered per unit time by a body moving in a circular path. It is represented by omega (ω).

2. Differentiate between time period and frequency.

Time Period	Frequency
Time for one revolution	Revolutions per second
Represented by T	Represented by ν
Unit = second	Unit = hertz

3. Write any two characteristics of centripetal acceleration.

1. It acts towards the centre of the circle.
2. Its direction changes continuously.

4. Why is centripetal acceleration not a constant vector?

Because its direction changes continuously during circular motion.

5. Explain relation between linear speed and angular speed.

The distance travelled in circular motion is related to angular displacement by:

s = Rθ

Dividing by time:

v = Rω

Thus linear speed equals radius multiplied by angular speed.

Long Answer Questions

1. Derive the relation between linear speed and angular speed.

Suppose an object moves through angular displacement Δθ in time Δt.

Distance travelled along arc:

Δs = RΔθ

Linear speed:

v = Δs / Δt

Substituting value of Δs:

v = RΔθ / Δt

Therefore:

v = Rω

Hence proved.

2. Derive centripetal acceleration in terms of angular speed.

We know:

a_c = v² / R

Using relation:

v = Rω

Substitute in formula:

a_c = (Rω)² / R

a_c = R²ω² / R

a_c = ω²R

Hence proved.

3. Derive speed in terms of frequency.

Distance covered in one revolution:

2πR

Time for one revolution:

T

Therefore:

v = 2πR / T

Since:

ν = 1 / T

Therefore:

v = 2πRν

Hence proved.

Assertion and Reason Questions

1. Assertion (A): Centripetal acceleration acts towards the centre.

Reason (R): Velocity changes continuously in circular motion.

Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

2. Assertion (A): Angular speed is constant in uniform circular motion.

Reason (R): Equal angles are covered in equal intervals of time.

Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

3. Assertion (A): Frequency and time period are directly proportional.

Reason (R): Frequency is number of revolutions per second.

Assertion is false but Reason is true.

Fill in the Blanks

1. Angular speed is represented by the symbol ________.

ω

2. The SI unit of frequency is ________.

Hertz

3. The time taken for one revolution is called ________.

time period

4. The relation between frequency and time period is ________.

ν = 1 / T

5. The formula of centripetal acceleration using angular speed is ________.

a_c = ω²R

Match the Following

Column A	Column B
1. Angular speed	a. Revolutions per second
2. Time period	b. rad/s
3. Frequency	c. Time for one revolution
4. Centripetal acceleration	d. Towards centre

Answers:

1 → b
2 → c
3 → a
4 → d

Statement-Based Questions

1. Identify true statements:

1. Angular speed is measured in rad/s.
2. Frequency is inverse of time period.
3. Velocity remains constant in circular motion.
4. Centripetal acceleration acts towards centre.

Statements 1, 2 and 4 are true.

2. Identify the false statement:

A. Frequency is measured in hertz
B. Time period is time for one revolution
C. Angular speed is measured in metres
D. Centripetal acceleration acts towards centre

C. Angular speed is measured in metres

Case Study Questions

A fan rotates with constant angular speed. The blades move in a circular path.

1. What type of motion is shown?

Uniform circular motion.

2. What is the direction of centripetal acceleration?

Towards the centre.

3. Write formula of angular speed.

ω = Δθ / Δt

4. Write relation between linear speed and angular speed.

v = Rω

5. What is the SI unit of angular speed?

rad/s