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Unit 2•Lesson 1 2 min read

Circular Motion and Universal Gravitation

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Learning Objectives

Define centripetal force and centripetal acceleration.
Apply the formula for centripetal force (Fc = mv²/r).
State Newton's Law of Universal Gravitation and use it to calculate the gravitational force between two masses.

Motion in a Circle

An object moving in a circle at a constant speed is still accelerating because its velocity is changing. Velocity is a vector, so a change in direction is a change in velocity.

Centripetal Acceleration and Force

Centripetal Acceleration (aₑ): The acceleration of an object moving in a circular path. It is always directed towards the center of the circle.
Formula: aₑ = v²/r
v is the object's speed.
r is the radius of the circular path.
Centripetal Force (Fₑ): According to Newton's Second Law (F=ma), if there is an acceleration, there must be a net force. The centripetal force is the net force required to keep an object moving in a circle. It is also directed towards the center of the circle.
Formula: Fₑ = maₑ = mv²/r
Important Note: Centripetal force is not a new, fundamental force of nature. It is simply the net force that is causing the circular motion. It could be tension in a string, gravity, or friction.

Example: For a planet orbiting the Sun, the centripetal force is provided by the force of gravity. For a car turning a corner, it is provided by the force of friction between the tires and the road.

Newton's Law of Universal Gravitation

Newton realized that the force that holds the Moon in its orbit around the Earth is the same type of force that makes an apple fall to the ground.

The Law: Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Formula: F₉ = G (m₁m₂) / r²
F₉ is the gravitational force.
G is the universal gravitational constant (≈ 6.674 x 10⁻¹¹ N·m²/kg²).
m₁ and m₂ are the masses of the two objects.
r is the distance between the centers of the two masses.

This is an inverse square law. If you double the distance between two objects, the gravitational force between them decreases by a factor of 2², which is 4.

Key Terms

Centripetal Force
A force that acts on a body moving in a circular path and is directed toward the center around which the body is moving.
Centripetal Acceleration
The acceleration of an object traveling in a circular path at a constant speed, directed towards the center of the circle.
Newton's Law of Universal Gravitation
A law stating that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Inverse Square Law
Any physical law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity.

Check Your Understanding

1

An object is moving in a circle at a constant speed. Is the object accelerating? Explain why or why not.

2

A 1000 kg car travels at 20 m/s around a circular track with a radius of 50 meters. What is the magnitude of the centripetal force required to keep the car on the track?

3

If the distance between two objects is tripled, by what factor does the gravitational force between them change?

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