Circular Motion: Forces, Acceleration, and Applications

Part 20 | Circular Motion

Introduction to Circular Motion

Circular motion occurs when an object moves along a circular path. Even if the object maintains a constant speed, its velocity is continuously changing direction, resulting in acceleration. This acceleration is always directed towards the center of the circle and is called centripetal acceleration.

• Constant Speed in Circular Path: The object’s speed remains unchanged, but its velocity vector changes direction.

• Equilibrium: An object moving in a circle at constant speed is not in equilibrium because there is a net force acting towards the center.

• Acceleration: The change in direction of velocity means the object is accelerating, even if its speed is constant.

Net Force and Acceleration in Circular Motion

The net force acting on an object in circular motion is called the centripetal force. This force is responsible for changing the direction of the velocity vector, keeping the object moving in a circle.

• Net Force Equation: $\sum F = ma$ For circular motion: $\sum F = F_c = ma_c$

• Centripetal Acceleration: $a_c = \frac{v^2}{r}$ where $v$ is the speed and $r$ is the radius of the circle.

• Centripetal Force: $F_c = m a_c = m \frac{v^2}{r}$

• Direction of Acceleration: Always points towards the center of the circle.

Principle of Circular Motion

If an object is moving in a circular path at a constant speed, its acceleration and net force point towards the center of the circle. These are called centripetal acceleration and centripetal force.

• Centripetal Force is not a new force: It is the name given to the net force causing circular motion, which can be the result of tension, gravity, friction, or normal force depending on the situation.

Examples of Centripetal Force

Situation	Centripetal Force
Planet orbiting the sun	Gravitational force ($F_g$)
Rock swung on a string	Tension force ($F_T$)
Person walking in a circle	Frictional force ($F_f$)
Roller coaster at top of loop	Sum of gravity and normal force ($F_g + F_N$)

Free-Body Diagrams in Circular Motion

Analyzing forces in circular motion often involves drawing a free-body diagram (FBD) to identify all forces acting on the object and determining which force provides the required centripetal force.

• Example: For a car moving in a circle, friction between the tires and the road provides the centripetal force.

Solving Circular Motion Problems

1. Draw a Free-Body Diagram (FBD).

2. Apply Newton’s Second Law: $\sum F = ma$.

3. Set the net force equal to the required centripetal force: $F_c = m \frac{v^2}{r}$.

4. Solve for the unknown (e.g., speed, radius, force).

Example Problem: Maximum Speed Without Skidding

A car travels around a circular path with a diameter of 248 meters. If the coefficient of static friction between the tires and the ground is $\mu_s = 0.93$, what is the fastest the car can move without skidding?

• Friction provides the centripetal force: $F_c = F_{fs}$

• Set up the equation: $\frac{mv^2}{r} = \mu_s F_N$

• Assume $F_N = mg$ (flat surface): $\frac{mv^2}{r} = \mu_s mg$

• Solve for $v$: $v^2 = \mu_s g r$ $v = \sqrt{\mu_s g r}$

Speed and Acceleration in Terms of Period

If an object travels with constant speed around a circle of radius $r$ and takes $T$ units of time to complete one circle (the period), the speed and acceleration can be expressed as:

• Speed: $v = \frac{2\pi r}{T}$

• Centripetal Acceleration: $a_c = \frac{v^2}{r} = \frac{4\pi^2 r}{T^2}$

• Centripetal Force: $F_c = m a_c = \frac{4\pi^2 r m}{T^2}$

Summary Table: Centripetal Quantities

Quantity	Formula
Speed	$v = \frac{2\pi r}{T}$
Centripetal Acceleration	$a_c = \frac{v^2}{r}$ or $\frac{4\pi^2 r}{T^2}$
Centripetal Force	$F_c = m \frac{v^2}{r}$ or $\frac{4\pi^2 r m}{T^2}$

Key Terms and Concepts

• Centripetal Acceleration ($a_c$): Acceleration directed towards the center of the circle.

• Centripetal Force ($F_c$): Net force required to keep an object moving in a circular path.

• Period ($T$): Time taken to complete one full circle.

• Velocity ($v$): Speed with direction; changes continuously in circular motion.

Additional info:

• In all cases, the centripetal force is provided by one or more physical forces (gravity, tension, friction, normal force).

• Centripetal force is not a separate force, but the name for the net force causing circular motion.