## Centripetal Acceleration

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# Centripetal Acceleration Calculator

### What is Centripetal Acceleration?

Consider a stone tied to a string and held by your hand. If you rotate by holding it firmly with your hand, the stone moves in a circular path, with a radius equal to the length of the string. The string held firmly by your hand exerts a force on the stone, which is termed as Centripetal Force.

As per the Newton’s second law, every object experiences an acceleration that is in the direction of the force acting on it. Here, the force is acting towards the center. Hence, the stone experiences an acceleration, termed as Centripetal acceleration, directed towards the center - which is your hand,.

Centripetal acceleration is an acceleration a body moving in circular path experiences due to the centripetal force.

If you leave the stone suddenly when it is moving in the circular path, it moves along the straight line that is tangential to that point at that instant. The velocity with which it leaves the circular path is called Tangential velocity ‘v’.

### What are the types of acceleration a body experiences when it is moving in a circular path?

An object moving in circular path experiences three types of accelerations:

- Angular Acceleration: α = Δω Δt
- Tangential Acceleration: at = rα in m s
^{2} - Centripetal Acceleration: ac = vt
^{2}r = rω^{2}.

While angular acceleration is an angular quantity, the rest two are linear quantities.

### What is the formula for Centripetal Acceleration?

For a body of mass ‘m’ moving in a circle of radius ‘r’ with a tangential velocity ‘v’, the centripetal acceleration is given by: ac = v^{2} / r

In terms of linear velocity, you can calculate the centripetal acceleration as ac= ω^{2}r, as we know that v=rω

The centripetal acceleration is also defined as the rate of change of tangential velocity: ac= Lim(Δv→0)[(Δvt/Δt)]

### Points to note about Centripetal acceleration

- Centripetal acceleration is always directed radially inwards towards the center of the circle in which it is moving.
- A body moving in circular path experiences both centripetal acceleration and angular acceleration. Thus the net acceleration that the body experiences is given by the root of the sum of the squares of both the accelerations.
- The moon that moves around the Earth is the best example of an object in a circular path. The gravity of the Earth causes centripetal acceleration on the Moon, that is always directed towards the Earth.
- If the angular velocity is kept constant, and you change the radius, centripetal acceleration will change; if radius decreases, centripetal acceleration increases and vice versa.
- Centripetal acceleration can change in two ways: either due to a huge change in centripetal velocity over a given time or due to the sudden change in centripetal velocity in a short span of time.
- The centripetal acceleration and tangential acceleration are always perpendicular to each other; because while the centripetal acceleration is directed towards the radius, tangential acceleration is in the direction of an instantaneous tangent. We know that radius and tangent are always perpendicular at a given point on the circle.

Note: An object moving in a circular path can have zero tangential acceleration, but never zero centripetal acceleration. If the body is having zero tangential acceleration, it means it is moving with constant angular velocity and that the angular acceleration is also zero.

But if a body moving in a circular path has zero centripetal acceleration, it means that its tangential velocity is not changing, which implies it is not moving in a circular at all.

The moon gravitates towards the earth and by the force of gravity is continually drawn off from a rectilinear motion and retained in its orbit.

- Isaac Newton

### Application of Centripetal Acceleration in day to day life

A device called Centrifuge works on the principle of centripetal acceleration. A centrifuge is a device that is used in labs and industries to separate particles of varying densities, mostly from liquids.

The centrifuge the container containing the liquid in rotation with high speed and applies a perpendicular force to the axis of the spin. As a result, the denser particles move outwards, while the lighter particles move towards the center.

This principle of centrifuge is commonly used in the following industries:

- Cream separator in dairies
- Waste processing units to separate suspended solids and liquids of varying densities.
- To simulate high gravity ambiance to train pilots and military personnel.
- Washing machines also use the principle of centrifuge.

Do you know?

To test the tolerance levels of astronauts, human centrifuges are designed with extremely large values of accelerations, that are often many times higher than the Earth’s gravity.

Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth’s gravity.

### How does CalculatorHut’s centripetal acceleration calculator help you?

Using CalculatorHut’s online free centripetal acceleration calculator, you can calculate any parameter related to the centripetal acceleration - radius, velocity, and centripetal acceleration. It is a very handy tool for students to check if their calculations are correct.

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