Centripetal force

The centripetal force is the force that causes an object to move in a circle, acting towards the centre of the circle. In the case of a satellite the centripetal force is gravity, in the case of an object at the end of a rope, the centripetal force is the tension of the rope.

It is important to understand right from the start that there is no 'default', 'natural' centripetal force. By default, objects tend to move in a straight line, as Newtonian mechanics teaches, away from the 'orbit', so in this context, by default there is only a centrifugal force at work. The centripetal force is being applied either by accident (meteors orbiting a planet) or artificially (satellites orbiting Earth, the object at the end of a rope etc). Therefore, the centrifugal force is a natural component of a circular movement, while the centripetal force is what we conventionally call the force keeping the object 'in orbit'.

Objects moving in a straight line with constant speed also have constant velocity. However an object moving in an arc with constant speed has a changing direction of motion. As velocity is a vector of speed and direction, a changing direction implies a changing velocity. The rate of this change in velocity is the centripetal acceleration. Differentiating the velocity vector gives the direction of this acceleration towards the centre of the circle. By Newton's second law of motion, as there is an acceleration there has to be a force in the direction of the acceleration. This is the centripetal force, and is equal to:

(where m is mass, v is velocity, r is radius of the circle, and the minus sign denotes that the vector points to the center of the circle and ω = v / r is the angular velocity )

Proof of law

Proving the formula is a trivial matter. Simply use a polar coordinate system, assume a constant radius, and take two derivatives.

Let r(t) be a vector that describes the position of a point mass as a function of time. Since we are assuming uniform circular motion, let r(t) = R·u_r, where R is a constant (the radius of the circle) and u_r is the unit vector pointing from the origin to the point mass. In terms of Cartesian unit vectors:

u_r = cos(θ)u_x + sin(θ)u_y

Note well: unlike in cartesian coordinates where the unit vectors are constants, in polar coordinates the direction of the unit vectors depend on the angle between the x-axis and the point being described; the angle θ.

So we take the first derivative to find velocity:

v = R du_r/dt = R dθ/dt u_θ

= R ω u_θ

where ω is the angular velocity (just a short way of writing dθ/dt), u_θ is the unit vector that is perpendicular to u_r that points in the direction of increasing θ. In cartesian terms: u_θ = -sin(θ) u_x + cos(θ) u_y

This result for the velocity is good because it matches out expectation that the velocity should the directed around the circle, and that the magnitude of the velocity should be ωR. Taking another derivative, we find that the acceleration, a is:

a = R( dω/dt u_θ - ω² u_r)

since the motion is uniform, the magnitude of v is constant, and thus there can be no dv/dt that points in the same direction as v. This fact simplifies the equation to:

a = -R ω² u_r

A simple substitution brings us to the equation above.