Centripetal force is the force that keeps an object moving in a circular path. It is always directed towards the center of the circle and is responsible for changing the direction of the object's velocity. The formula for centripetal force is given by F = (mv^2)/r, where F is the centripetal force, m is the mass of the object, v is the velocity of the object, and r is the radius of the circular path.

Explanation of the Formula:

1. Centripetal Force:
- Centripetal force is a force that acts towards the center of the circular path, keeping the object moving in that path.
- It is responsible for changing the direction of the object's velocity, but not its speed.

2. Mass and Velocity of the Object:
- The formula includes the mass (m) of the object moving in a circular path.
- It also includes the velocity (v) of the object, which is the speed at which the object is moving.

3. Radius of the Circular Path:
- The radius (r) of the circular path is the distance from the center of the circle to the object.
- It determines the size of the circular path and affects the magnitude of the centripetal force.

4. Deriving the Formula:
- The centripetal force is derived from Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
- In the case of circular motion, the acceleration is centripetal acceleration (a = v^2/r), which is directed towards the center of the circle.
- Substituting the centripetal acceleration into Newton's second law, we get F = m(v^2/r), which is the formula for centripetal force.

Summary:
The formula for centripetal force, F = (mv^2)/r, describes the force required to keep an object moving in a circular path. It takes into account the mass of the object, the velocity at which it is moving, and the radius of the circular path. Understanding this formula helps in determining the necessary force to maintain circular motion and provides insights into various applications, such as satellite orbits, amusement park rides, and car racing.

Centripetal force (from Latin centrum, "center" and petere, "to seek"[1]) is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre".[2] In Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits.

One common example involving centripetal force is the case in which a body moves with uniform speed along a circular path. The centripetal force is directed at right angles to the motion and also along the radius towards the centre of the circular path.[3][4] The mathematical description was derived in 1659 by the Dutch physicist Christiaan Huygens.