# Centripetal Acceleration And Tangential Acceleration: 7 Facts

This article talks about centripetal acceleration and tangential acceleration. Acceleration as we all know is the rate at which the velocity of an object increases.

For an object accelerating in a circular motion, the acceleration will have two main components. One acting towards the centre and other acting along the tangent of the circular path. Let us discuss about these two types of accelerations in detail.

## What is acceleration?

Acceleration can be defined as the rate with which a moving object change its velocity. We can estimate the time of arrival or project the value of velocity after a certain time interval using the value of acceleration.

Image credits: Wikipedia

Mathematically, acceleration can be given as,

a = dv/dt

where,

dv is the instantaneous change in velocity

dt is the instantaneous change in time

## What is centripetal acceleration?

Centripetal acceleration, as the name suggests, is that component of acceleration that acts towards the centre of the circular path.

Mathematically, the centripetal acceleration is given as-

$Ac= v^{2}/r$

where,

V is the tangential velocity of the object

r is the radius of rotation

## What is tangential acceleration?

Tangential acceleration as the name suggests is that value of acceleration that acts towards along the tangent of the circular path.

Mathematically, the tangential acceleration can be given as-

[latex]a_{t}=r\alpha {/latex]

where,

at is the tangential acceleration

Alpha is the angular acceleration

r is the radius of rotation

## Is centripetal acceleration and tangential acceleration same?

No. The centripetal acceleration and tangential acceleration are two different components of acceleration. The major difference between them is the direction.

Centripetal acceleration acts perpendicular to the tangential acceleration which is long the centre of the circular path. Whereas, the tangential acceleration acts along the tangent of the circular path. Both these accelerations can be found by resolving the acceleration vector.

## How are centripetal acceleration and tangential acceleration related?

Centripetal acceleration comes into play when there is a change in the direction of tangential velocity of the object whereas the tangential acceleration changes only when there is a change in tangential velocity of the object.

Circular motion cannot take place without centripetal acceleration but it is possible to have a circular motion with zero tangential acceleration.

## Does centripetal acceleration affect tangential acceleration?

The centripetal acceleration does not bring any change in tangential acceleration. Centripetal acceleration can cause a change in objects direction without changing its tangential velocity.

If tangential velocity is constant then the tangential acceleration will be zero. Hence we can say that centripetal acceleration does not have any direct influence on tangential acceleration.

## Is tangential acceleration equal to centripetal acceleration?

Both the types of acceleration have different meaning but then can be same in values.

Centripetal acceleration acts towards the centre of the circular path whereas the tangential acceleration acts along the tangent of the circular path. Their magnitudes can be equal depending on the values of velocity.

## The angle between centripetal acceleration and tangential acceleration is?

The vectors of centripetal acceleration and tangential acceleration are perpendicular to each other.

This is because one acts towards the centre and other along the tangent of the circular path. This way the angle between the vectors becomes 90 degrees.

## Centripetal and tangential acceleration example

1. Earth revolving around sun– When planets revolve around their respective suns, they possess both tangential acceleration and centripetal acceleration. According to Kepler, the velocity of revolution increases as the planet comes nearer to the sun. The planets follow an elliptical orbit around the sun.
2. Stone and thread example– When we tie a stone with thread and start revolving the the thread around one finger, then the stone will have both centripetal and tangential acceleration if we keep changing the radius of rotation or change the speed of rotation.
3. Satellite in orbit– Satellites are injected into an orbit using a rocket. These satellites travel very fast and orbit around the planet after every fixed interval. However there are some course corrections to be made, during course corrections the magnitude of centripetal acceleration and tangential acceleration changes. Even when the satellite is sitting inside a rocket, it keeps on accelerating till the orbital velocity is reached.
4. Moon in orbit– Moon in orbit is a similar example to revolution of planets around sun, it possesses both tangential as well as centripetal acceleration.

## Practice problem

Consider the following given data

Velocity: 5 m/s

Velocity after 5 secs: 10m/s

Radius of rotation after 5s= 10m

Time period of rotation at 1 s: 5 s

Time period of rotation after 5 sec (at t=5 secs) : 10 s

Find the centripetal acceleration and tangential acceleration from the data given above

Solution:

The formula for centripetal acceleration is already discussed in the above section, after substituting the values in the formula, we get centripetal acceleration as 5m/s2

Now let us calculate the value of tangential acceleration

The formula of tangential acceleration has a term called angular acceleration, alpha. We can find the value of alpha using the formula given below-

[latex}\alpha = (\omega _{2}-\omega _{1})/(t2-t1){/latex}

where,

alpha is the angular acceleration

Omega is the angular velocity

t is the time interval

Substituting the values in the formula given above-

we get the value of angular acceleration as 0.2 rad/s2

from the above formula we get the value of tangential acceleration at t=5s as 2 m/s2

## Conclusion

In this article we have discussed about both tangential acceleration and centripetal acceleration. We conclude that centripetal acceleration acts towards the centre of the circular motion whereas the tangential acceleration acts towards the tangent of the circular motion. We also conclude that centripetal acceleration depends on the change in direction of tangential velocity whereas the value of tangential acceleration depends on the value of tangential velocity.

Abhishek

I, Abhishek Khambhata, have pursued Btech in Mechanical Engineering. Throughout four years of my engineering, I have designed and flown unmanned aerial vehicles. My forte is fluid mechanics and thermal engineering. My fourth year project was based on performance enhancement of unmanned aerial vehicles using solar technology. I would like to connect with like minded people.