Circular Motion: Exploring Centripetal Force and Acceleration

What Is Circular Motion?

Circular motion occurs when an object moves in a circular path due to a centripetal force acting toward the centre.

Key Equations in Circular Motion

Angular Velocity (\(\omega\))

The rate of change of angular displacement:

\[ \omega = \frac{\theta}{t} \]

Where:

• \(\omega\): Angular velocity (rad/s)
• \(\theta\): Angular displacement (radians)
• \(t\): Time (s)

Centripetal Force (\(F_c\))

The force keeping an object in circular motion:

\[ F_c = \frac{mv^2}{r} \]

Where:

• \(F_c\): Centripetal force (N)
• \(m\): Mass (kg)
• \(v\): Tangential velocity (m/s)
• \(r\): Radius of the circle (m)

Centripetal Acceleration (\(a_c\))

The acceleration toward the center of the circle:

\[ a_c = \frac{v^2}{r} \]

Real-Life Applications of Circular Motion

Transportation

• Banking roads help cars maintain circular motion safely

Space Science

• Satellites use gravitational centripetal force to stay in orbit

Engineering

• Centrifuges rely on circular motion for separating substances

Example Problem

A car of mass \(1,000 \, \text{kg}\) travels at \(20 \, \text{m/s}\) around a curve with a radius of \(50 \, \text{m}\). Find the centripetal force.

1. Formula:
   \[ F_c = \frac{mv^2}{r} \]
2. Substitute Values:
   \[ F_c = \frac{1,000 \times 20^2}{50} = 8,000 \, \text{N} \]

Common Mistakes in Circular Motion Problems

1. Forgetting to square the velocity in centripetal force calculations
2. Mixing up angular and tangential velocity
3. Misinterpreting the direction of centripetal force (always toward the center)

Practice Questions

1. A \(500 \, \text{g}\) mass travels at \(10 \, \text{m/s}\) on a circular path of radius \(2 \, \text{m}\). Find the centripetal acceleration.
2. Explain why satellites stay in orbit due to circular motion.
3. Describe one application of centripetal force in transportation.