Definition of Centripetal Force

Centripetal force is a force acting on an object moving along a curved path. The direction of this force is always towards the center of the curve and is what keeps the object on that path, preventing it from continuing its movement in a straight line.

Curvilinear movement and centripetal force

Suppose we have an object moving along a circular path. To describe the curvilinear movement of this body, angular and linear variables are used. Angular variables are those that describe the movement of the object in terms of the angle that it “sweeps” along its path. On the other hand, linear variables are those that use its position with respect to the point of rotation and its speed in the tangential direction of the curve.

The centripetal acceleration \({a_c}\) experienced by an object moving in a trajectory circular with a tangential speed \(v\) and at a distance \(r\) from the point of rotation will be given by:

\({a_c} = \frac{{{v^2}}}{r}\)

Centripetal acceleration is a linear variable that is used to describe curvilinear movement and is directed towards the center of the curved path. On the other hand, the angular velocity ω of the object, that is, the rate of change of the swept angle (in radians) per unit of time, is given by:

\(\omega = \frac{v}{r}\)

Or, we can solve for \(v\):

\(v = \omega r\)

This is the relationship that exists between linear velocity and angular velocity. If we plug this into the expression for centripetal acceleration we get:

\({a_c} = {\omega ^2}r\)

Newton's second law tells us that the acceleration of a body is directly proportional to the force applied to it and inversely proportional to its mass. Or, in its best known form:

\(F = ma\)

Where \(F\) is the force, \(m\) is the mass of the object and \(a\) is the acceleration. In the case of curvilinear motion, if there is a centripetal acceleration there must also be a force centripetal \({F_c}\) that acts on the body of mass \(m\) and that causes the centripetal acceleration \({a_c}\), is say:

\({F_c} = m{a_c}\)

Substituting the previous expressions for the centripetal acceleration we obtain that:

\({F_c} = \frac{{m{v^2}}}{r} = m{\omega ^2}r\)

The centripetal force is directed towards the center of the curvilinear path and is responsible for constantly changing the direction in which the object moves to keep it moving curved.

Gravity as a centripetal force and Kepler's Third Law

Kepler's third law of planetary motion states that the square of the orbital period, that is, the time The time it takes for a planet to complete one orbit around the Sun is proportional to the cube of the semimajor axis of the orbit. That is:

\({T^2} = C{r^3}\)

Where \(T\) is the orbital period \(C\), it is a constant and \(r\) is the semimajor axis, or the maximum distance between the planet and the Sun throughout its orbit..

For simplicity, consider a planet of mass \(m\) moving along a circular orbit around the Sun, although this analysis can be extended to the case of an elliptical orbit and obtain the same result. The force that keeps the planet in its orbit is gravity, which will be:

\({F_g} = \frac{{G{M_S}m}}{{{r^2}}}\)

Where \({F_g}\) is the force of gravity, \({M_S}\) is the mass of the Sun, \(G\) is the universal gravitation constant and \(r\) is the distance between the planet and the sun. However, if the planet moves along a circular orbit, it experiences a centripetal force \({F_c}\) that keeps it on said trajectory and that in terms of the angular velocity \(\omega \) will be given by:

\({F_c} = m{\omega ^2}r\)

The curious thing is that in this case gravity is that centripetal force that keeps the planet in its orbit, in a few words \({F_g} = {F_c}\), therefore, we can say that:

\(\frac{{G{M_S}m}}{{{r^2}}} = m{\omega ^2}r\)

Which we can simplify as:

\(G{M_S} = {\omega ^2}{r^3}\)

The angular velocity is related to the orbital period in the following way:

\(\omega = \frac{{2\pi }}{T}\)

Substituting this into the previous equation we obtain that:

\(G{M_S} = \frac{{4{\pi ^2}}}{{{T^2}}}{r^3}\)

Rearranging the terms we finally obtain that:

\({T^2} = \frac{{4{\pi ^2}}}{{G{M_S}}}{r^3}\)

The latter is precisely Kepler's Third Law that we presented previously and if we compare the proportionality constant it would be \(C = 4{\pi ^2}/G{M_S}\).

What about centrifugal force?

It is more common for this type of movement to speak of “centrifugal force” instead of centripetal force. Above all, because it is what we apparently feel when we experience this. However, centrifugal force is a fictitious force resulting from inertia.

Let's imagine that we are riding in a car that is traveling at a certain speed and suddenly brakes. When this happens we will feel a force that pushes us forward, however, this apparent force that we feel is the inertia of our own body that wants to maintain its state of movement.

In the case of a curvilinear movement, the centrifugal force is the inertia of the body that wants to maintain its rectilinear movement but which is subject to a centripetal force that keeps it on the curved path.

References

David Halliday, Robert Resnick & Jearl Walker. (2011). Fundamentals of Physics. United States: John Wiley & Sons, Inc.