derivation of centripetal acceleration - Overview, Structure, Properties & Uses

Q: Centripetal acceleration formula derivation.

Derivation: F=mv 2 /r(using centripetal force) Also, F=ma(using newton’s 2 nd law) We can rewrite it as a=F/m Substituting the value in the above equation, a=mv 2 /r/m Simplifying it, we get a=v 2 /r

Q: Centripetal acceleration formula proof.

Proof: F=mv 2 /r(using centripetal force) Also, F=ma(using newton’s 2 nd law) We can rewrite it as a=Fm Substituting the value in the above equation, a=mv 2 /r/m Simplifying it, we get a=v 2 /r

Derivation of Centripetal Acceleration - Detailed Guide

Edited By Team Careers360 | Updated on May 26, 2022 05:06 PM IST

In this article we will be studying centripetal acceleration is, centripetal acceleration formula, centripetal acceleration derivation, centripetal force, define centripetal acceleration, direction of centripetal acceleration and centrifugal acceleration formula.

Define centripetal acceleration and derive an expression for it.

Centripetal acceleration is the acceleration of a body that is travelling across a circular path. When a body undergoes a circular motion, its direction constantly changes and thus its velocity changes (velocity is a vector quantity) which produces an acceleration. The centripetal acceleration ac is given by the square of speed v divided by the distance r;

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Centripetal acceleration Formula:

a_c = v²/r

This is the required Centripetal acceleration Formula.

Centripetal acceleration unit: metre per second squared. The force causing this acceleration is also directed towards the centre of the circle and is named centripetal force.

Also read -

Centripetal Acceleration Derivation.

Consider a body of mass ‘m’ moving on the circumference of a circle of radius ‘r’ with a velocity ‘v’. A force F is then applied on the body. And this force is given by

F = ma

, the centripetal acceleration equation is given by a=v^2/r.

Where, a= acceleration which is given by the rate of change of velocity with respect to time.

In △OAB and △PQR,

then Δv / AB=v/r

Clearly, AB=vΔt
⟹Δv / vΔt=v/r
⟹ Δv / Δt=v²/r

⟹a=v²/r [Expression for centripetal acceleration]

Thus, the centripetal acceleration equation is given by a=v²/r or centripetal acceleration is v 2 r.

Direction of centripetal acceleration(& force) is towards the centre of the circle.

What is the direction of centripetal acceleration?

The direction of centripetal acceleration and also force is towards the centre of the circle.

Centripetal Force

It is the force that acts on a body undergoing circular motion and is directed towards the centre of the rotation(or the circle).

Related Topics Link,

Centripetal Force Derivation.

Centripetal force is the net force causing uniform circular motion.

According to Newton’s laws of motion, force F=ma where m is mass and a is acceleration.

The acceleration in uniform circular motion is centripetal acceleration. a_c=v²/r or a_c=rω²where v is linear velocity, ⍵ is angular velocity, and r is radius of curvature.

Then centripetal force formula of linear velocity is given by:

F_c=m v²/r

Then centripetal force formula in terms of angular velocity is given by:

F_c=mrω²

It is the mrw2 formula/centripetal force formula angular velocity

[v=ωr in vector form /v rw in vector form can be written as $v=\omega \times\vec r$ ]

Therefore, the expression for centripetal force is given by F = mv²/r. Or we can say that mv²/r is the formula of centripetal acceleration.

It is also known as angular centripetal force/derivation of centrifugal force class 11

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Centrifugal force formula derivation.

Centrifugal force derivation

NOTE:

For a particle in circular motion, the centripetal acceleration is: a_c=v²/r or a_c=rω²
Expression of centripetal acceleration
mω²r formula is for Force.
Centrifugal acceleration formula/ Centrifugal acceleration equation.
F= mω²r = mv²/r

Derive v= r ω

Also read :

Let us consider a body rotating about an axis that is passing through O point and also perpendicular to the plane. Let us suppose, P be the position of a particle inside the body. If the body rotates an angle 0 in a time ‘t’, the particle at which is at P reaches P′
∴ Linear displacement PP′=rθ
∴Linear velocity, v=PP'/t =rθ/t
But θ/t=ω, the angular velocity
∴Linear velocity , v=rω

Derive an expression for centripetal acceleration in uniform circular motion.

F=mv²/r(using centripetal force)

Also, F=ma(using newton’s 2^nd law)

We can rewrite it as a=F/m

Substituting the value in the above equation,

a=mv²/r/m

Simplifying it, we get

a=v²/r

( As we know v=rω)

a=ω²r

Also check-

Frequently Asked Question (FAQs)

1. Centripetal acceleration formula derivation.

Derivation:

F=mv²/r(using centripetal force)

Also, F=ma(using newton’s 2^nd law)

We can rewrite it as a=F/m

Substituting the value in the above equation,

a=mv²/r/m

Simplifying it, we get

a=v²/r

2. Centripetal acceleration formula proof.

Proof:

F=mv²/r(using centripetal force)

Also, F=ma(using newton’s 2^nd law)

We can rewrite it as a=Fm

Substituting the value in the above equation,

a=mv²/r/m

Simplifying it, we get

a=v²/r

3. Define centripetal acceleration derive an expression for it./What is centripetal acceleration derive an expression for it.

Centripetal acceleration is the acceleration of a body that is traveling across a circular path. When a body undergoes a circular motion, its direction constantly changes, and thus its velocity changes (velocity is a vector quantity) which produce an acceleration.

Derivation:

F=mv²/r(using centripetal force)

Also, F=ma(using newton’s 2^nd law)

We can rewrite it as a=Fm

Substituting the value in the above equation,

a=mv²/r/m

Simplifying it, we get

a=v²/r

4. Derive an expression for centripetal force class 11. / Derivation of centripetal force class 11.

Done above

5. Define centripetal acceleration class 11 and also write the expression for centripetal acceleration class 11.

Derivation: