Centripetal force is defined as the force acting on a body that is moving in a circular path that is directed toward the center around which the body moves. The term comes from the Latin words *centrum* for center and *petere*, meaning "to seek". Centripetal force may be considered the center-seeking force. Its direction is orthogonal to the motion of the body in the direction toward the center of curvature of the body's path. Centripetal force alters the direction of an object's motion without changing its speed.

### Key Takeaways: Centripetal Force

- Centripetal force is the force on a body moving in a circle that points inward toward the point around which the object moves.
- The force in the opposite direction, pointing outward from the center of rotation, is called centrifugal force.
- For a rotating body, the centripetal and centrifugal force are equal in magnitude, but opposite in direction.

## Difference Between Centripetal and Centrifugal Force

While centripetal force acts to draw a body toward the center of the point of rotation, the centrifugal force (center-fleeing force) pushes away from the center. According to Newton's First Law, "a body at rest will remain at rest, while a body in motion will remain in motion unless acted upon by an external force". The centripetal force allows a body to follow a circular path without flying off at a tangent by continuously acting at a right angle to the path.

The *centripetal force requirement* is a consequence of Newton's Second Law, which says an object being accelerated undergoes a net force, with the direction of the net force the same as the direction of the acceleration. For an object moving in a circle, the centripetal force must be present to counter the centrifugal force.

From the standpoint of a stationary object on the rotating frame of reference (e.g., a seat on a swing), the centripetal and centrifugal are equal in magnitude, but opposite in direction. The centripetal force acts on the body in motion, while the centrifugal force does not. For this reason, centrifugal force is sometimes called a "virtual" force.

## How to Calculate Centripetal Force

The mathematical representation of centripetal force was derived by Dutch physicist Christiaan Huygens in 1659. For a body following a circular path at constant speed, the radius of the circle (r) equals the mass of the body (m) times the square of the velocity (v) divided by the centripetal force (F):

r = mv^{2}/F

The equation may be rearranged to solve for centripetal force:

F = mv^{2}/r

An important point you should note from the equation is that centripetal force is proportional to the square of velocity. This means doubling the speed of an object needs four times the centripetal force to keep the object moving in a circle. A practical example of this is seen when taking a sharp curve with an automobile. Here, friction is the only force keeping the vehicle's tires on the road. Increasing speed greatly increases force, so a skid becomes more likely.

Also note the centripetal force calculation assumes no additional forces are acting on the object.

## Centripetal Acceleration Formula

Another common calculation is centripetal acceleration, which is the change in velocity divided by the change in time. Acceleration is the square of velocity divided by the radius of the circle:

Δv/Δt = a = v^{2}/r

## Practical Applications of Centripetal Force

- The classic example of centripetal force is the case of an object being swung on a rope. Here, the tension on the rope supplies the centripetal "pull" force.
- Centripetal force is the "push" force in the case of a Wall of Death motorcycle rider.
- Centripetal force is used for laboratory centrifuges. Here, particles that are suspended in a liquid are separated from the liquid by accelerating tubes oriented so the heavier particles (i.e., objects of higher mass) are pulled toward the bottom of the tubes. While centrifuges commonly separate solids from liquids, they may also fractionate liquids, as in blood samples, or separate components of gases. Gas centrifuges are used to separate the heavier isotope uranium-238 from the lighter isotope uranium-235. The heavier isotope is drawn toward the outside of a spinning cylinder. The heavy fraction is tapped and sent to another centrifuge. The process is repeated until the gas is sufficiently "enriched".
- A liquid mirror telescope (LMT) may be made by rotating a reflective liquid metal, such as mercury. The mirror surface assumes a paraboloid shape because the centripetal force depends on the square of the velocity. Because of this, the height of the spinning liquid metal is proportional to the square of its distance from the center. The interesting shape assumed by spinning liquids may be observed by spinning a bucket of water at a constant rate.