Example 3.9 Uniform Circular Motion Solution Step by Step

  Angular Speed and Linear Speed Numerical with Solution

Dr.Sanjaykumar pawar

Uniform circular motion showing angular speed, linear speed and centripetal acceleration for an insect moving in a circular path.

Internal Links

Introduction to Uniform Circular Motion

Angular Velocity Formula and Numerical Problems

Linear Velocity in Circular Motion Explained

Centripetal Force and Centripetal Acceleration

Motion in a Plane Class 11 Notes

NCERT Physics Class 11 Solved Examples

Circular Motion Important Formulas PDF

Difference Between Speed and Velocity

Uniform vs Non-Uniform Circular Motion

Class 11 Physics Chapter-Wise Numerical Solutions

Example 3.9 – Uniform Circular Motion

Example 3.9 An insect trapped in a circular groove of radius 12 cm moves along the groove steadily and completes 7 revolutions in 100 s. (a) What is the angular speed, and the linear speed of the motion? (b) Is the acceleration vector a constant vector ? What is its magnitude

Given Data

• Radius of groove, R = 12 cm
• Number of revolutions = 7
• Total time taken = 100 s

(a) Find Angular Speed (ω) and Linear Speed (v)

Step 1: Find Time Period (T)

Time period is the time taken to complete one revolution.

T = Total Time / Number of Revolutions

T = 100 / 7

T = 14.3 s

Step 2: Calculate Angular Speed (ω)

ω = 2π / T

ω = (2 × 3.14) / 14.3

ω = 0.44 rad s⁻¹

Angular Speed (ω) = 0.44 rad s⁻¹

Step 3: Calculate Linear Speed (v)

v = ωR

v = 0.44 × 12

v = 5.28 cm s⁻¹

v ≈ 5.3 cm s⁻¹

Linear Speed (v) = 5.3 cm s⁻¹

(b) Is the Acceleration Vector Constant?

Step 1: Direction of Velocity

In circular motion, velocity always acts along the tangent to the circle. As the insect moves, the tangent changes continuously. Therefore, the direction of velocity changes continuously.

Step 2: Direction of Acceleration

The acceleration is always directed towards the centre of the circle. This is called centripetal acceleration.

As the insect moves around the circle, the direction towards the centre keeps changing. Therefore, the acceleration vector is not constant.

Acceleration Vector = Not Constant

Step 3: Find Magnitude of Acceleration

a = ω²R

a = (0.44)² × 12

a = 0.1936 × 12

a = 2.32 cm s⁻²

a ≈ 2.3 cm s⁻²

Magnitude of Acceleration = 2.3 cm s⁻²

Final Answers

(a) Angular Speed = 0.44 rad s⁻¹
(a) Linear Speed = 5.3 cm s⁻¹

(b) Acceleration vector is NOT constant because its direction changes continuously.
Magnitude of acceleration = 2.3 cm s⁻²

Quick Exam Notes

• Velocity is always tangent to the circle.
• Centripetal acceleration is always towards the centre.
• Speed remains constant in uniform circular motion.
• Velocity changes because its direction changes.
• Acceleration vector is not constant.
• Magnitude of acceleration remains constant.