Unit 2.9: Circular Motion

Introduction

Circular motion occurs when an object moves along a curved or circular path. Unlike linear motion, where velocity points in a straight line, circular motion requires a continuously changing velocity direction. This change is caused by centripetal force, which always points toward the center of the circle.

Key Concepts

Centripetal Acceleration: The acceleration directed toward the center of a circular path:
\( a_c = \frac{v^2}{r} \)
Centripetal Force: The net force required to keep an object in circular motion:
\( F_c = m \frac{v^2}{r} \)
Period (\(T\)): The time taken for one complete revolution:
\( T = \frac{2\pi r}{v} \)
Frequency (\(f\)): The number of revolutions per second:
\( f = \frac{1}{T} \)

Tip: Centripetal force is not a separate force. It results from other forces (gravity, tension, friction) acting toward the center.

Types of Circular Motion

Uniform Circular Motion: The speed remains constant, but the direction of velocity changes.
Non-Uniform Circular Motion: The speed changes, meaning there is tangential acceleration in addition to centripetal acceleration.

Example: A satellite orbiting Earth moves in uniform circular motion, while a car speeding up in a curved exit ramp experiences non-uniform circular motion.

Graphical Representations

Understanding motion in a circular path often requires vector diagrams and free-body diagrams:

Velocity Vectors: Always tangent to the circular path.
Acceleration Vectors: Always directed toward the center.
Force Diagrams: Show forces such as gravity, normal force, or tension providing the required centripetal force.

Exam Strategy: When analyzing forces, identify what provides the necessary centripetal force. For example, friction keeps a car turning on a curve, while tension keeps a ball moving in a circle on a string.

Mathematical Routines

Circular motion problems often involve:

Using \( F_c = m v^2 / r \) to solve for mass, velocity, radius, or force.
Applying Newton’s Second Law in a circular path: \( \sum F = m a_c \).
Determining the period of motion using \( T = \frac{2\pi r}{v} \).

Important: If a car loses traction on a curve, it does not fly outward due to “centrifugal force.” Instead, it continues in a straight line due to inertia.

Practice Activities

Activity 1: Centripetal Acceleration

A 1,200 kg car rounds a 50 m radius curve at 20 m/s. What is the required centripetal force?

Activity 2: Identifying Forces

A 5 kg mass is whirled in a horizontal circle on a string of 1.5 m length at a speed of 4 m/s. Identify what force is providing the centripetal acceleration and calculate its value.

Activity 3: Period of Motion

A satellite orbits a planet in a circular path with a radius of 10,000 km and a speed of 5,000 m/s. Calculate its period of motion.

Summary & Exam Preparation Tips

Circular motion is a key concept in Newtonian mechanics. Key takeaways:

Circular motion requires a centripetal force directed toward the center.
Acceleration in uniform circular motion is given by \( a_c = v^2 / r \).
The period and frequency describe how fast an object completes a revolution.
Circular motion problems often involve applying Newton’s Second Law in radial direction.

Mastering circular motion will help with related topics such as orbits, banked curves, and rotational dynamics.