We experienced it every day while driving, but it's somehow hard to grasp: with our calculator, the centripetal force won't be a mystery anymore.

Read this article to learn more about this physics topic. We are sure that it's going to satisfy your needs, whether you are curious or you can't really bend your mind around the concept. Here you will learn:

  • What is the centripetal force?
  • How to calculate the centripetal acceleration and the centripetal force: equations for a circular motion;
  • How to apply the centripetal force formula to homework and examples;
  • Another formula for the centripetal acceleration and force; and
  • Why we don't talk about centrifugal force?

What is the centripetal force?

First thing: imagine a body moving in a linear fashion. How does it change direction? The answer is by the application of a force. A change in direction equals a change in the intensity of one or more of the cartesian components of the velocity of the body. This change is due to Newton's first law of motion, which states that "a body remains at rest unless an external force acts on it".

Take a car moving on a highway. After a long straight stretch, finally, a bend appears. To turn (change direction), a force attracts the car in the direction of the center of the turn. The force in question is the centripetal force.

The centripetal force allows your car to turn. If you are wondering which force actually enables the turn, think of friction. By turning your wheels, you change the component of the friction in the desired direction.

How do I calculate the centripetal force? Centripetal force and centripetal acceleration equation

The formula for the centripetal force is:

F=mv2rF = \frac{m\cdot v^2}{r}

Where:

  • FF is the centripetal force;
  • mm is the mass;
  • vv is the speed; and
  • rr is the curvature radius.

Invoking Newton's second law, we can pass from the centripetal force equation to find the centripetal acceleration:

a=Fm=v2ra = \frac{F}{m}=\frac{v^2}{r}

Now you know how to calculate the centripetal acceleration. Take a quick look at the measurement units of the involved quantities:

  • The force is measured in newtons in the metric system, and in pounds-force or poundals in the imperial system;
  • The acceleration is either measured in m/s2\text{m}/\text{s}^2 or ft/s2\text{ft}/\text{s}^2;
  • The mass, rather straightforwardly uses kilograms or pounds, and the radius meters or feet (alternatively kilometers and miles).

The centripetal acceleration equation in action: examples of centripetal force

Take a turn with curvature radius r=300 mr=300\ \text{m}. At which speed do you have to drive if you want to experience an 1 g1\ \text{g} acceleration during the turn?

Insert the data in our centripetal force calculator, and... yeah, you have to go supersonic!

v=ar=9.81 ms2300 m=542.5 ms=1, ⁣953 kmh\begin{align*} v& = \sqrt{a\cdot r}=\sqrt{9.81\ \frac{\text{m}}{\text{s}^2} \cdot 300\ \text{m}}\\ &=542.5\ \frac{\text{m}}{\text{s}} = 1,\!953\ \frac{\text{km}}{\text{h}} \end{align*}

That's clearly too much. At a reasonable speed of 110 km/h110\ \text{km}/\text{h}, the experienced acceleration would be less than the one felt while accelerating from 00 to 100100 in 6 seconds.

Other ways to calculate the centripetal force

Since the centripetal force is defined for circular motions, we can identify other formulas for the centripetal acceleration that use quantities directly connected to that kind of motion.

The following equation for the centripetal force uses the angular velocity:

F=mω2rF = m\cdot\omega^2\cdot r

The corresponding formula for the centripetal acceleration is:

a=ω2ra = \omega^2 \cdot r

ω\omega, the angular velocity, is measured in radians, or degrees per second.

What about the centrifugal force?

Never mistake centrifugal and centripetal force: for starters, the former is not a real force. The apparent force that pushes you on the outside of the turn in a fast-moving car is a consequence of inertia. Your body is trying to maintain its current state of motion: a straight line and the car pressing on you that gives the impression of a force pushing you outside. The centrifugal force is a consequence of the fact that once you start a turn, you are not in an inertial reference frame anymore.

Davide Borchia
Equation for centrifugal force
Mass (m)
lb
Radius (r)
ft
Tangential velocity (v)
ft/s
Angular velocity (ω)
rad/s
Centripetal force (F)
lbf
Centripetal acceleration (a)
m/s²
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