# Derivation of an expression for centripetal acceleration

• Sophieg
In summary, the conversation is about deriving an expression for the centripetal acceleration of a satellite orbiting Earth, with the given expression for its velocity being v= root of GM/r. The individual is unsure of which equations to use and how to equate them, but eventually realizes that the answer is GM/r^2.
Sophieg

## Homework Statement

I have derived the expression for the velocity of the satellite v= root of GM/r however I'm struggling to derive an expression for the centripetal acceleration of a satellite orbiting Earth.

## The Attempt at a Solution

I'm not entirely sure which equations to relate or equate. To find the velocity I equated the force between two masses and the centripetal force and rearranged.

update: I think I've worked it out.. I equated the equation for centripetal force F=mv^2/r (of the planet) to the force between the two masses (planet and the satellite) to get:
- Gm1m2/r^2 = mv^2/r ... canceled a mass from either side and obtained Gm/r^2=v^2/r and since v^2/r = centripetal acceleration then the answer is GM/r^2?
Is this correct?

Hi sophleg. Welcome to PF!

Your answer is correct but I am a bit puzzled by how you determined that ##v = \sqrt {GM/r}## before working out the centripetal acceleration.

AM

Hello Andrew,

Thank you!

I equated the force between two masses F=Gm1m2/r^2 = centripetal force F=mv^2/r and rearranged for v!

## What is centripetal acceleration?

Centripetal acceleration is the acceleration that a body experiences when it is moving in a circular path. It is always directed towards the center of the circle.

## What is the equation for centripetal acceleration?

The equation for centripetal acceleration is a = v²/r, where a is the centripetal acceleration, v is the velocity of the object, and r is the radius of the circle.

## How is the expression for centripetal acceleration derived?

The expression for centripetal acceleration is derived using the principles of circular motion and Newton's second law of motion. By equating the net force acting on a body in circular motion to its mass times its centripetal acceleration, we can derive the formula a = v²/r.

## What is the relationship between centripetal acceleration and tangential acceleration?

Centripetal acceleration and tangential acceleration are both components of the total acceleration of a body in circular motion. Centripetal acceleration is directed towards the center of the circle, while tangential acceleration is directed tangentially to the circle. They are related by the equation a² = ac² + at².

## What are some real-life examples of centripetal acceleration?

Some examples of centripetal acceleration in everyday life include the motion of a car around a curve, a roller coaster on a loop, and a satellite orbiting the Earth. Any object that moves in a circular path experiences centripetal acceleration.

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