Centripetal and Centrifugal Force

Have you ever heard the terms "centripetal force" and "centrifugal force" used interchangeably? This wording can be confusing when trying to understand the forces that are acting on an object experiencing circular motion. In this article, we will discuss the difference between centripetal force and centrifugal force. We will also go over the equations that describe circular motion and go through some examples of objects experiencing circular motion.

Explanation of Centripetal Force and Centrifugal Force

In order to understand centripetal force and centrifugal force, we need to understand the dynamics of an object moving in a circular motion. Let's consider a ball on a string moving in a circle, as shown below. If the ball is moving with a constant velocity around the circle, it is experiencing uniform circular motion.

As shown in the image above, the velocity of the ball is always in a direction tangent to the circle and thus the direction of the velocity vector is constantly changing. This causes the ball to experience acceleration perpendicular to the velocity vector, and thus the acceleration vector is always pointing to the center of the circle, as shown in the image. This radial acceleration is known as the centripetal acceleration.

We know from Newton's second law that there must be a force acting upon an object if it has an acceleration. This law is described by the equation$F_{net}=ma$, where$F_{net}$is the sum of the forces acting on the object,$m$is the mass of the object, and$a$is the acceleration. In our case, we are considering the centripetal acceleration of a ball in uniform circular motion, so we will label our centripetal acceleration as$a_c$so that$F_{net}=m{a}_c$.

So, what is this force? From an observer's reference frame, if we ignore gravity there is only one force acting on the ball on the string: the tension force from the string holding it. The tension provides the radial force necessary to keep it in a circular motion. The net radial force acting on an object that keeps it moving in a circle is the centripetal force. The centripetal force vector points in the same direction as the acceleration vector according to Newton's second law.

What if we consider the ball's reference frame though? From the ball's reference frame, the ball is at rest while everything around it is in motion. If we only consider the centripetal force from the tension in the string in the ball's reference frame, the ball should be moving to its left towards the center of the circle. To fix this, we use a "fictitious force" which is the centrifugal force. The centrifugal force is directed radially outwards and has the same magnitude as the centripetal force.

Difference Between Centripetal and Centrifugal Force

Centripetal force is the component of force acting on a rotating body which is directed towards the axis of rotation. Centrifugal force is a pseudo-force within the reference frame of the rotating body which acts outwards along the radius of rotation, pushing the body away from the axis of rotation.

The difference between the centripetal force and the centrifugal force is that while the centripetal force applies to any reference frame, the centrifugal force is only applicable in a rotating reference frame. The observer's reference frame is called an inertial reference frame, which means that the law of inertia, Newton's first law, holds true. A rotating reference frame is a non-inertial reference frame because the law of inertia does not hold true.

If we consider a ball placed on a rotating, frictionless platform, the ball will be pushed off of the platform. This makes sense for an observer watching the ball because the ball had initial velocity from the rotating platform. But from the ball's rotating reference frame, the ball begins to move radially outward from the platform without any forces acting on it. Thus in a non-inertial reference frame, Newton's laws of motion do not hold true. We can, however, use Newton's laws of motion in a non-inertial reference frame if we introduce "fictitious forces" such as centrifugal force. Fictitious forces make calculations in non-inertial reference frames easier since they allow us to use Newton's laws of motion.

Developing Formulas for Centripetal Force and Centrifugal Force

From the image shown above, we can draw two similar triangles for the change in velocity$∆v$and the change in distance$∆x$. Since the triangles are similar triangles, the ratio of similar sides of the triangles is equal and gives us:

$\frac{∆x}{R}=\frac{∆v}{v_1}$

$\frac{\left|\overrightarrow{∆v}\right|}{\overrightarrow{v_1}}=\frac{∆x}{R}\Rightarrow \left|\overrightarrow{∆v}\right|=\frac{∆x}{R}\overrightarrow{v_1}$

Now let's think about the average acceleration. Acceleration is defined as the change in velocity divided by the change in time. Thus, we can write:

$$\begin{gathered} a=\frac{∆v}{∆t} \\ =\frac{∆x}{R}\frac{v_1}{∆t} \\ =\frac{v_1}{R}\frac{∆x}{∆t} \end{gathered}$$

If we consider very small changes in distance and time, we can take the limit as the change in time approaches zero. As the change in time approaches zero, the change in distance over the change in time approaches$v_1$.

$\underset{∆t\to 0}{lim}\frac{∆x}{∆t}\Rightarrow v_1$

Now we can write the acceleration as:

$$\begin{gathered} a=\frac{v_1}{R}\underset{∆t\to 0}{lim}\frac{∆x}{∆t} \\ =\frac{v_1}{R}\left(v_1\right) \end{gathered}$$

Since we have taken the limit as the change in time approaches zero, we can drop the subscripts. We now arrive at the equation for the centripetal acceleration:

$a_c=\frac{v^2}{R}$

The centripetal force points in the same direction as the centripetal acceleration, to the center of the circle.

The centrifugal force,$F_{cf}$, used only in a non-inertial reference frame, has the same magnitude as the centripetal force:

$F_{cf}=m\frac{v^2}{R}$

Though the centripetal force and the centrifugal force are equal in magnitude, they point in opposite directions. The centrifugal force is directed radially outwards.

Example of Centripetal Force and Centrifugal Force

$\theta ={\sin }^{-1}\left(\frac{R}{L}\right)$

$$\begin{gathered} T=\frac{F_g}{\cos \theta } \\ =\frac{mg}{\cos \theta } \end{gathered}$$

$$\begin{gathered} T_x=T\sin \theta \\ =\left(\frac{mg}{\cos \theta }\right)\sin \theta \\ =mg\tan \theta \end{gathered}$$

Applications of Centripetal Force and Centrifugal Force

A common application that involves the centripetal force and centrifugal force is a car driving around a flat curve at a constant speed. In an inertial reference frame, the forces acting on the car are gravity, the normal force, and friction, which keeps the car from sliding. The centripetal force is supplied by the friction, keeping the car moving in a circle. A passenger in the car could argue that they feel an outward force pushing them into the door as the car goes around the curve, but in an inertial reference frame, there is no centrifugal force pushing outward. What the passenger is feeling comes from the law of inertia. The passenger has an initial velocity from the car in motion and wants to move in a straight line, but since the car is rounding a curve, the side of the car keeps the passenger from moving in a straight line. If we consider a non-inertial reference frame from the passenger's perspective, the passenger and car are at rest while everything else is moving. The outward force the passenger feels can be called the centrifugal force in their reference frame, but we need to remember that it is not a real force since there is no object applying it.