Physics: Circular Motion and Centripetal Force

Physics Formula, Circular Motion and Centripetal Force - Formula Quest Mania

Introduction to Physics Formula Circular Motion and Centripetal Force

Circular motion is a foundational concept in classical mechanics. It refers to the motion of an object along the circumference of a circle or a circular arc. This type of motion can be observed in everyday life — from the rotation of the Earth around the Sun, to the spinning wheels of a car, or a child twirling a toy on a string.

Uniform vs Non-uniform Circular Motion

In uniform circular motion, an object moves along a circular path with constant speed. However, the direction of its velocity vector is continuously changing, indicating a constant acceleration directed toward the center.

In non-uniform circular motion, both the speed and direction of the object change. This leads to tangential acceleration in addition to the centripetal acceleration.

Key Quantities in Circular Motion

1. Angular Displacement (\(\theta\))

Angular displacement is the angle through which an object moves on a circular path. It is measured in radians.

2. Angular Velocity (\(\omega\))

The angular velocity is defined as:

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

3. Linear Velocity (v)

Linear velocity is given by:

\[ v = r \cdot \omega \]

4. Centripetal Acceleration (\(a_c\))

The formula for centripetal acceleration is:

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

What is Centripetal Force?

Centripetal force is the force that keeps an object moving in a circular path. It always acts perpendicular to the motion of the object and toward the center of the circle.

Formula for Centripetal Force

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

Relation to Newton’s Second Law

From Newton’s second law:

\[ F = ma \]

In the case of circular motion, the relevant acceleration is the centripetal acceleration:

\[ F_c = m \cdot \frac{v^2}{r} \]

Real-World Examples of Centripetal Force

1. A Satellite Orbiting Earth

Satellites in low Earth orbit experience gravitational force from the Earth that acts as the centripetal force, keeping them in orbit. Their tangential speed balances the pull of gravity.

2. Roller Coaster Loops

When a roller coaster passes through a vertical loop, the centripetal force required to keep the cars on track is provided by the normal force and the gravitational pull, especially at the top of the loop.

3. Banked Roads

Highways and racetracks use banked curves to assist vehicles in making turns. Banking reduces the reliance on friction by allowing a component of the normal force to act as centripetal force.

Example Problems

Example 1: Find Centripetal Force

Given: Mass = 5 kg, Speed = 10 m/s, Radius = 2 m

Solution:

\[ F_c = \frac{mv^2}{r} = \frac{5 \cdot 100}{2} = 250\,N \]

Example 2: Angular Velocity from Linear Velocity

Given: v = 12 m/s, r = 3 m

\[ \omega = \frac{v}{r} = \frac{12}{3} = 4\, \text{rad/s} \]

Example 3: Velocity Needed for a Turn

Problem: A car of mass 1000 kg must make a turn of radius 50 m. What is the maximum velocity if the available frictional force is 5000 N?

\[ F_c = \frac{mv^2}{r} \Rightarrow v = \sqrt{\frac{F_c \cdot r}{m}} = \sqrt{\frac{5000 \cdot 50}{1000}} = \sqrt{250} \approx 15.81\, \text{m/s} \]

Applications in Engineering and Space Science

1. Particle Accelerators

In circular particle accelerators like cyclotrons, magnetic fields supply the centripetal force needed to keep charged particles in a circular path. Their speed increases with each pass through the accelerator.

2. Space Missions

Centripetal dynamics are essential in designing orbits for space missions. Engineers calculate orbital speed using:

\[ v = \sqrt{\frac{GM}{r}} \]

Where \(G\) is the gravitational constant, \(M\) is the mass of the central body (e.g., Earth), and \(r\) is the orbital radius.

3. Washing Machines

Washing machines use centripetal force during the spin cycle. Water is forced out of the clothes because there is no inward centripetal force acting on the water, so it moves outward through the holes of the drum.

Lab Experiments for Circular Motion

1. Rubber Stopper and String Setup

A rubber stopper attached to a string is swung in a circle. By adjusting the radius and speed, students can measure the force on the string and verify the centripetal force formula.

2. Rotating Platforms

A lab with a rotating platform can be used to measure angular velocity and compare tangential and centripetal acceleration using motion sensors.

Advanced Concepts

1. Non-Inertial Frames and Pseudo Forces

In a rotating frame of reference, fictitious forces such as the centrifugal force appear to act on objects. Though not real, they help explain observations from the rotating perspective.

2. Centripetal vs Centrifugal Force

Centripetal force is real and always points inward toward the center. Centrifugal force is a pseudo force that appears to act outward when viewed from a rotating frame.

3. Conservation of Angular Momentum

When an ice skater pulls in their arms, they reduce their moment of inertia and spin faster to conserve angular momentum:

\[ L = I \cdot \omega \]

Summary of Formulas

\( \omega = \frac{\theta}{t} \) — Angular velocity
\( v = r \cdot \omega \) — Linear velocity
\( a_c = \frac{v^2}{r} = r \cdot \omega^2 \) — Centripetal acceleration
\( F_c = \frac{mv^2}{r} = m \cdot r \cdot \omega^2 \) — Centripetal force

Conclusion

Understanding circular motion and centripetal force is critical in analyzing motion in rotating systems. From amusement parks to planetary orbits and engineering systems, these concepts provide the foundation for safe and efficient design. Using precise mathematical relationships, scientists and engineers can model, predict, and control systems involving rotation.

Practice Questions

A 3 kg object moves in a circle with a radius of 2 m at 8 m/s. What is the centripetal force?
If the radius of a circle is doubled and the speed is constant, how does centripetal force change?
How is angular velocity affected if the time to complete a revolution is halved?

Formula Quest Mania