Derivation of Centripetal Acceleration - Detailed Guide

What is Centripetal Acceleration

Centripetal acceleration is the acceleration of a body moving in a circular path, which is always directed towards the centre of the circle. It is responsible for continuously changing the direction of the velocity of the body, thereby keeping it in circular motion.

Even if the speed of the body remains constant, its velocity changes due to the change in direction, and hence acceleration is produced. This acceleration is called centripetal acceleration.

The magnitude of centripetal acceleration depends on the speed of the body and the radius of the circular path. It is given by:

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

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 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 triangles $\triangle O A B$ and $\triangle P Q R$, then:

$
\Delta v_{A B}=\frac{v}{r}
$
Clearly, from the diagram (not shown here), the arc length $A B=v \Delta t$.
Now, we have:

$
\frac{\Delta v}{v \Delta t}=\frac{v}{r}
$
Simplifying this:

$
\frac{\Delta v}{\Delta t}=\frac{v^2}{r}
$
Thus, the centripetal acceleration equation is:

$
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

What is the direction of centripetal acceleration?

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

Uses of Centripetal Acceleration

Centripetal acceleration is widely used to understand and explain circular motion in physics and real life. Some important uses are:

• It helps in analyzing the motion of planets and satellites revolving around a central body.
• It is used in designing roads and circular tracks, where proper banking ensures safe turning of vehicles.
• It explains the working of devices like centrifuges, which separate substances based on density.
• It is used in studying the motion of objects tied to a string and rotated in a circle.
• It helps in understanding the motion of electrons in atoms and other circular paths in physics.

Difference Between Centripetal and Tangential Acceleration

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Solved Examples of Centripetal Acceleration

Q.1 A car moves in a circular path of radius $\mathbf{2 0 ~ m}$ with a speed of $\mathbf{1 0 ~ m} / \mathbf{s}$.
Find its centripetal acceleration.
Solution:
Using formula,

$
\begin{gathered}
a_c=\frac{v^2}{r} \\
a_c=\frac{(10)^2}{20}=\frac{100}{20}=5 \mathrm{~m} / \mathrm{s}^2
\end{gathered}
$

Centripetal acceleration $=5 \mathrm{~m} / \mathrm{s}^2$.

Q.2 A stone tied to a string rotates in a horizontal circle of radius 0.5 m with an angular speed of 6 rad/s. Calculate centripetal acceleration.

Solution:
Using formula,

$
\begin{gathered}
a_c=\omega^2 r \\
a_c=(6)^2 \times 0.5=36 \times 0.5=18 \mathrm{~m} / \mathrm{s}^2
\end{gathered}
$

Centripetal acceleration $=18 \mathrm{~m} / \mathrm{s}^2$.

Q.3 A fan blade rotates with an angular speed of $\mathbf{2 0 ~ r a d} / \mathrm{s}$. If the edge of the blade is $\mathbf{0 . 3 ~ m}$ from the centre, find the centripetal acceleration of the tip.

Solution:

$
\begin{gathered}
a_c=\omega^2 r \\
a_c=(20)^2 \times 0.3=400 \times 0.3=120 \mathrm{~m} / \mathrm{s}^2
\end{gathered}
$

Centripetal acceleration $=120 \mathrm{~m} / \mathrm{s}^2$.

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