Circular Motion - Notes for Quick Revision 📚

Circular motion is a type of motion in which an object moves along a circular path. This topic is crucial in physics as it helps explain the dynamics of objects in circular trajectories, from planets orbiting the Sun to vehicles taking a turn on a curved road.

This article provides detailed notes on circular motion, covering definitions, types, formulas, and applications, along with problem-solving strategies.

What is Circular Motion?

Definition: Circular motion refers to the motion of an object along a circular path.

Key Features:

• The object’s distance from the center of the circle remains constant.
• It involves both tangential and radial components of motion.

Types of Circular Motion

1. Uniform Circular Motion (UCM)

Definition: Motion in which an object moves along a circular path with constant speed.

Characteristics:

• Speed remains constant, but velocity changes due to the continuous change in direction.
• Acceleration is directed towards the center of the circle (centripetal acceleration).

2. Non-Uniform Circular Motion

Definition: Motion in which an object moves along a circular path with a varying speed.

Characteristics:

• Both the magnitude and direction of velocity change.
• Centripetal acceleration is present, and tangential acceleration accounts for the change in speed.

Key Terms in Circular Motion

1. Angular Displacement ($\theta$)

• The angle subtended by the radius vector at the center during the motion.
• Unit: Radian ($\text{rad}$).

2. Angular Velocity ($\omega$)

• The rate of change of angular displacement.
• Formula: $$\omega = \frac{d\theta}{dt}$$
• Unit: $\text{rad/s}$.

3. Angular Acceleration ($\alpha$)

• The rate of change of angular velocity.
• Formula: $$\alpha = \frac{d\omega}{dt}$$
• Unit: $\text{rad/s}^2$.

4. Centripetal Force ($F_c$)

• The inward force required to keep an object moving in a circular path.
• Formula: $$F_c = \frac{mv^2}{r} = m\omega^2 r$$
• Unit: Newton ($\text{N}$).

5. Centripetal Acceleration ($a_c$)

• The acceleration directed towards the center of the circular path.
• Formula: $$a_c = \frac{v^2}{r} = \omega^2 r$$

6. Tangential Velocity ($v$)

• The linear velocity of the object along the tangent to the circular path.
• Formula: $$v = r\omega$$
• Unit: $\text{m/s}$.

Equations in Circular Motion

1. Relation Between Linear and Angular Quantities

• Tangential velocity: $$v = r\omega$$
• Tangential acceleration: $$a_t = r\alpha$$

2. Time Period ($T$)

• The time taken by an object to complete one full revolution.
• Formula: $$T = \frac{2\pi}{\omega}$$
• Unit: Seconds ($\text{s}$).

3. Frequency ($f$)

• The number of revolutions per second.
• Formula: $$f = \frac{1}{T}$$
• Unit: Hertz ($\text{Hz}$).

Forces in Circular Motion

Centripetal Force:

• Required to keep an object in circular motion.
• Examples: Gravitational force (planets), frictional force (cars on curves).

Centrifugal Force:

• A pseudo force experienced in a rotating reference frame, acting outward from the center.

Applications of Circular Motion

Planets and Satellites:

• Orbital motion of planets and satellites involves gravitational centripetal force.

Vehicles on Curves:

• Friction provides the centripetal force required for turning.

Rotating Machines:

• Fans and turbines rely on uniform circular motion principles.

Amusement Rides:

• Rides like Ferris wheels and roller coasters operate using circular motion dynamics.

Banking of Roads

When a vehicle moves along a curved path, the road is banked at an angle to provide the necessary centripetal force.

Key Formula:

For a banked road with no friction:

$$\theta = \tan^{-1} \left( \frac{v^2}{rg} \right)$$

Where:

• $\theta$: Banking angle.
• $v$: Velocity of the vehicle.
• $r$: Radius of the curve.
• $g$: Acceleration due to gravity.

Vertical Circular Motion

In vertical circular motion, the forces acting on the object vary at different points of the path.

At the Topmost Point:

• Centripetal force: $$F_c=T+mg$$

At the Bottommost Point:

• Centripetal force: $$F_c=T-mg$$

Graphical Representation

Angular Velocity vs Time:

• For uniform circular motion: Constant.
• For non-uniform circular motion: Linearly increasing or decreasing.

Centripetal Force vs Radius:

• Inversely proportional ($F_c\propto \frac{1}{r}$).

Problem-Solving Tips

Understand the Motion Type:

Identify whether the motion is uniform or non-uniform.

Draw a Free-Body Diagram (FBD):

Represent forces acting on the object, especially centripetal force.

Use Appropriate Formulas:

Select equations based on angular or linear quantities.

Consider Reference Frames:

Account for pseudo-forces if analyzing motion in a non-inertial frame.

FAQs About Circular Motion

What is the difference between centripetal and centrifugal force?

• Centripetal force acts towards the center of the circular path.
• Centrifugal force is a pseudo force acting outward in a rotating reference frame.

Can an object move in a circular path without a force?

No, a centripetal force is required to change the direction of motion continuously.

What is the significance of banking of roads?

Banking helps reduce reliance on friction to provide the necessary centripetal force, ensuring safer turns.

Why does an object experience acceleration in uniform circular motion?

Even though the speed is constant, the continuous change in direction results in centripetal acceleration.

Circular motion is a fundamental concept that explains a wide range of natural and technological phenomena. By understanding the dynamics of uniform and non-uniform circular motion, along with forces like centripetal and centrifugal, students can solve problems in mechanics with confidence. These notes offer a comprehensive guide for academic and competitive exam preparation.

Explore related topics:

• Kinematics
• Rotational Motion and Moment of Inertia
• Angular Momentum in Rotational Motion