Centripetal Force Formula

Centripetal force is the force acting on any item moving along a curved route. This force always travels in the same direction as the radius of the curve. When we discuss circular motion, we usually use centripetal force examples. It is the most basic form of nonlinear movement. The radius of the curvature in this situation is, of course, the radius of the circle. 

Centripetal Force Formula

The centripetal force formula may be written as:

\[F = \frac{m \times v^{2}}{r}\]

F: Centripetal force;

m: Mass of the object;

v: Velocity

r: Curvature's (circle's) radius.

According to Newton's second law,

\[a = \frac{v^{2}}{r}\],is the centripetal acceleration formula.

To visualise what centripetal force definition is all about, look at the diagram below:

(Image Will be Updated Soon)

Centripetal Force Equation

We may also replace the velocity with the angular velocity (\[\omega\]) in the centripetal force equation.

\[ F_{c} = m \times \omega^{2} \times r\]

According to Newton's second law,

\[a = \frac{\omega^{2}}{r}\], is the centripetal acceleration formula.

Centripetal Force Formula Derivation

Starting with Newton's \[2^{nd}\] law :

\[a = \frac{F}{m}\]

and then equating this to the centripetal acceleration equation,

\[\frac{v^{2}}{r} = \frac{F}{m}\]

The centripetal force is denoted by \[ F_{c}\].

\[ F_{c} = \frac{ m \times v^{2}}{r}\]

And the expression for centripetal acceleration is, this force is always directed towards the centre of the circular path. Equivalently, if (\[\omega\]) is the angular velocity then because \[v = \omega r\],

\[ F_{c} = m \times \omega ^{2} \times r\]

Centripetal Force Dimensional Formula

Let discuss the expression for centripetal force,

The Centripetal force dimensional formula for the centripetal force is calculated using the fundamental units which are defined exclusively.

The centripetal force formula is given below:

\[ F_{c} = \frac{ m \times v^{2}}{r}\]

Now, the dimension for each quantity is derived as follows:

\[\Rightarrow~m~ =~(M)\]

\[\Rightarrow~r~ =~(L)\]

\[\Rightarrow~v~ =~(LT^{-1})\]

Here, on substituting each dimension on the formula, we get,

\[F = \frac{(M)(LT^{-1})^{2}}{L}\]

\[F = \frac{(ML^{2}T^{-2})}{L}\]

\[F = MLT^{-2}\]

Hence, the dimensional formula of centripetal force is \[MLT^{-2}\]

Centripetal Force Units

• The Newton, N, is the SI unit of centripetal force.

• The poundal (pd1) is the imperial unit of centripetal force.

• The pound-force (lbf) is the English engineering unit of centripetal force.

• The CGS unit is the dyne.

Centrifugal Force

At first glance, there may seem to be no difference between the centripetal and centrifugal force formula, as the centrifugal force equation is identical to the centripetal force equation.

Hence centrifugal force formula is \[ F_{c} = \frac{ m \times v^{2}}{r}\],

Let's look at both diagrams with the comparison of centripetal force vs. centrifugal force:

(Image Will be Updated Soon)

Solved Examples:

Ex.1. A truck of 5,050 Kg is travelling at 50.0 m/s covers a curve of the radius of 200 m. calculate the centripetal force.

Solution

The given parameters are

mass = 5,050 Kg

radius = 200 m

Velocity = 50.0 m/s

Substitute the values in the given centripetal force formula

\[ F_{c} = \frac{ m \times v^{2}}{r}\]

\[ F_{c} = \frac{ 5050 \times 50.0^{2}}{200}\]

\[ F_{c} = 63125~N\]

Ex.2. Let's find the velocity of an object that travels around the circle with radius r = 7 ft when the centrifugal force equals 5.6 pdl and its mass is 4 lb.

Solution:

mass = 4 lb

radius = 7 ft

Centrifugal force = 5.6 pdl

Velocity =?

By using centrifugal force equation, we get,

\[ F_{c} = \frac{ m \times v^{2}}{r}\]

Rearranging the terms,

\[v^{2} = \frac{ F_{c} \times r}{m}\]

\[v^{2} = \frac{ 5.6 \times 7}{4}\]

\[v^{2} = 9.8\]

\[v = \sqrt{9.8}\]

\[v = 3.13 m/s\]

Ex.3. A travelling bus of 6,870 Kg is travelling at 40.0 m/s covers a curve of the radius of 300 m. calculate the centripetal force.

Solution: The given parameters are

mass = 6,870 Kg

radius = 300 m

Velocity = 40.0 m/s

Substitute the values in the given centripetal force formula

\[F_{c} = \frac{m \times v^{2}}{r}\]

\[F_{c} = \frac{6870 \times 40.0^{2}}{300}\]

\[F_{c} = 36,640 N\]