Derivation of Centripetal Acceleration - Detailed Guide

Imagine driving around a curve and feeling a pull outward—that's centripetal acceleration in action, a key concept in circular motion. It helps explain how forces work on a vehicle during turns. Understanding this can make driving safer, especially when navigating curves at higher speeds. In this report, we explore the role of centripetal acceleration, factors that affect it like speed and friction, and practical tips for handling turns.

This concept, covered in Class 11 physics under circular motion, is essential for board exams and competitive exams like JEE Main, NEET, BITSAT, and others. Between 2013 and 2023, five questions on this topic have appeared in JEE Main. In this article we will be studying centripetal acceleration, centripetal acceleration formula, Derivation of centripetal acceleration class 11, centripetal force, centripetal acceleration, direction of centripetal acceleration and centrifugal acceleration formula.

Define Centripetal Acceleration

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".

Centripetal Acceleration Formula:

Centripetal acceleration $a_c=\frac{v^2}{r}$

This is the required Centripetal acceleration Formula.

Centripetal acceleration unit: metre per second squared (m/s²).

Note: The force causing this acceleration is also directed towards the centre of the circle and is named centripetal force.

Also read -

• NCERT Solutions for Class 11 Physics
• NCERT Solutions for Class 12 Physics
• NCERT Solutions for All Subjects

Centripetal Acceleration Derivation

Now let's start Derivation of centripetal acceleration class 11

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 to the body. And this force is given by

F = ma

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

Consider the $\mathrm{△}OAB$ and $\mathrm{△}PQR$, then $\frac{\mathrm{\Delta }v}{AB}=\frac{v}{r}$

Clearly from above image, A B = v Δ t

$\frac{\mathrm{\Delta }v}{v\mathrm{\Delta }t}=\frac{v}{r}$

$\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}=\frac{v^2}{r}$

$a=\frac{v^2}{r}$

Thus, the Expression for centripetal acceleration Class 11 is given by $a=\frac{v^2}{r}$

Note: The direction of centripetal acceleration(& force) is towards the centre of the circle.

Now, let's move to another part of this article which is Centripetal Force

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).

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 = Mass of body

And a = Acceleration of the body

The acceleration in uniform circular motion is centripetal acceleration.

$a_c=\frac{v^2}{r}$

Here,

v is the linear velocity of the object.

r is the radius of the circular path

Using Angular Velocity ( ω ) ,

$a_c=r{\omega }^2$

Here,
( ω ) is the angular velocity of the object.

r is the radius of the circular path.

Then centripetal force formula of linear velocity is given by:

$F_c=\frac{m{v}^2}{r}$

Here,

F_c is the centripetal force.

m is the mass of the object.

v is the linear velocity.

r is the radius of the circular path.

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

$F_c=mr{\omega }^2$

Here:

F_c is the centripetal force.

m is the mass of the object.

r is the radius of the circular path.

ω is the angular velocity.

Therefore, the expression for centripetal force is given by

$F_c=mr{\omega }^2$

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

NCERT Physics Notes :

• NCERT Notes class 11 physics
• NCERT Notes class 12 physics
• NCERT Notes for all subject

Centrifugal Force Formula Derivation

$\text{Centrifugal Force}$

$F=m\times \frac{v^2}{r}$

$F=m\times \frac{(r\omega {)}^2}{r}\text{or}F=m\times {\omega }^{2}r$

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 :

• NCERT solutions for Class 11 Physics Chapter 3 Motion in a straight line
• NCERT Exemplar Class 11 Physics Solutions Chapter 3 Motion in a straight line
• NCERT notes Class 11 Physics Chapter 3 Motion in a straight line

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′

$\begin{array}{c}P{P}^{\prime }=r\theta \\ v=\frac{P{P}^{\prime }}{t}=\frac{r\theta }{t}\\ \frac{\theta }{t}=\omega \\ v=r\omega \end{array}$

Derive an expression for centripetal acceleration in a 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,

$a=\frac{F_c}{m}=\frac{m{v}^2}{rm}$

$\text{Simplify:}$

$a=\frac{v^2}{r}$

$\text{Using the relationship}v=r\omega :$

$v^2=(r\omega {)}^2=r^2{\omega }^2$

$\text{Substitute}{v}^2\text{into the acceleration formula:}$

$a=\frac{r^2{\omega }^2}{r}$

$\text{Simplify:}$

$a=r{\omega }^2$

So, the centripetal acceleration can be expressed as:

$a=\frac{v^2}{r}=r{\omega }^2$

Centripetal Force vs centrifugal Force

Centripetal Force:

Definition: Centripetal force is the force that acts on an object moving in a circular path, directed towards the center of the circle. It keeps the object moving in a curved trajectory rather than in a straight line.

Nature: Real force that acts on the object.

Direction: Always directed towards the centre of the circular path.

Formulae: $F_c=\frac{m{v}^2}{r}$

Examples:

1. The tension in a string when swinging a ball in a circular motion.

2. The gravitational force acting on planets keeps them in orbit.

Centrifugal Force:

Definition: Centrifugal force is a pseudo or fictitious force that appears to act on an object when it is observed from a rotating reference frame. It seems to push the object away from the centre of rotation.

Nature: Not a real force; it is an apparent force observed in a non-inertial (rotating) frame of reference.

Direction: Directed away from the centre of the circular path, opposite to the direction of centripetal force.

Formula: $F_{\text{centrifugal}}=-\frac{m{v}^2}{r}=-mr{\omega }^2$

Examples:

1. The sensation of being "thrown" outward when a car makes a sharp turn.

2. The apparent force felt by passengers in a spinning amusement park ride.

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Problem Solving Strategy

1. Identify the plane of circular motion.

2. Locate the centre of rotation and calculate the radius.

3. Make F.B.D.

4. Resolve force along the radial direction and along the direction perpendicular to it.

5. The net force along the radial direction is mass times the radial acceleration i.e. $m\left(\frac{V^2}{R}\right)$ or $m\left(\omega^2 R\right)$ Centripetal force $\left(\frac{\mathrm{mV}^2}{\mathrm{R}}\right)$ is no separate force like Tension, Weight, Spring force, Normal reaction, Friction ect. In fact anyone of these or their combination may play a role of centripetal force.

Also check-

• NCERT Exemplar Class 11th Physics Solutions
• NCERT Exemplar Class 12th Physics Solutions
• NCERT Exemplar Solutions for All Subjects