How do you derive the formula for centripetal force?

The formula for centripetal force is derived from Newton's second law of motion and the definition of acceleration.

To understand the derivation of the formula for centripetal force, we need to start with 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, which is directed towards the centre of the circle.

Centripetal acceleration is given by the formula a = v²/r, where v is the velocity of the object and r is the radius of the circle. This formula comes from the definition of acceleration as the rate of change of velocity. In circular motion, the direction of the velocity is constantly changing (even if its magnitude, the speed, remains constant), which means there is always an acceleration towards the centre of the circle.

Substituting the formula for centripetal acceleration into Newton's second law gives F = m(v²/r). This is the formula for centripetal force, which tells us that the force required to keep an object moving in a circle is proportional to the mass of the object, the square of its velocity, and inversely proportional to the radius of the circle.

It's important to note that centripetal force is not a new kind of force, but rather the name given to the force or combination of forces that act towards the centre of the circle to keep an object moving in circular motion. This could be tension in a string, gravitational force, friction, or any other force depending on the situation. The direction of the centripetal force is always towards the centre of the circle, which is why it's often described as a 'centre-seeking' force.

IB Physics Tutor Summary: The formula for centripetal force, F = m(v²/r), is derived by combining Newton's second law (F = ma) with the formula for centripetal acceleration (a = v²/r). This shows that the force keeping an object moving in a circle depends on its mass, speed squared, and the circle's radius. Centripetal force isn't a new type, but a force acting towards the circle's centre.