# Centripetal Force

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7.2Centripetal Force

In these revision notes for Centripetal Force, we cover the following key points:

• The meaning of centripetal force
• The situations in which centripetal force exists
• What causes the centripetal force?
• Equation of centripetal force
• How does weight change when a vertical centripetal force is present in the system?
• How to calculate the total acceleration of the system in a non-uniform rotational motion

## Centripetal Force Revision Notes

From Newton's Second Law of Motion we known that where there is an acceleration, there is also a force causing it. Centripetal acceleration makes no exception to this rule as well. Therefore, it is obvious there must be a force causing the centripetal acceleration. This force is known as Centripetal Force, FC and it is the force responsible for keeping an object in rotation. It is in the same direction of centripetal acceleration, i.e. towards the centre of circle.

As centripetal acceleration point towards the centre of curvature in the direction of circle's radius is often referred as "radial acceleration, ar".

The equation of Centripetal Force based on the Newton's Second Law of Motion, is:

FC = m × aC

where m is the object's mass. In scalar form, the above equation is written as:

FC = m × v2/r

When centripetal force stops acting, the object moves linearly in the direction of the tangent line to the circle it had at the last moment when FC was still present in the system.

Centripetal force is not a specific force in itself; many forces such as frictional, gravitational, tension force etc. act as centripetal forces in certain conditions.

Weight is not always numerically equal to the gravitational force. As an example in this regards we can mention situations when an object is moving according a vertical rotational path. In this case, the equation of weight is

W = Fg - Fextra (vertical)

and since this extra vertical force is nothing else but the centripetal force, we can write

W = Fg - FC

When an object is at the lowest point of a circular trajectory we have W > Fg and when it is at the highest point of a circular trajectory, then W < Fg.

When an object is rotating uniformly around a fixed point, the only acceleration present in the system is the centripetal acceleration, aC. However, when the object's speed increases or decreases during rotation, there also exists another acceleration due to speeding up or slowing down. Since the direction of this acceleration complies with the direction of velocity (it is a kind of instantaneous acceleration whose direction is collinear with the vector of instantaneous velocity), we call it "tangential acceleration, at" and it is perpendicular to the centripetal acceleration aC.

The acceleration caused by the change in the magnitude of velocity (here the tangential acceleration at) is calculated by the equation

at = v - v0/∆t

where v and v0 are the magnitudes of final and initial velocities respectively and Δt is the time interval this process occurs.

On the other hand, we have the centripetal (radial) acceleration whose formula is

aC = v2/r

As a result, since aC and at are perpendicular, we can use the Pythagorean Theorem to calculate the magnitude of the total acceleration a in a non-uniform rotational motion:

a = √a2t + a2C

The direction of total acceleration is obtained by applying the parallelogram rule for the addition of vectors.

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