Centripetal Force: Definition, Formula & Examples

When an object is traveling in a circular path, centripetal force is what keeps it fixed in that path. Learn more about this force, how it is calculated and examples of its occurrence. A quiz is provided to test your learning.
Definition of Centripetal Force
Do you remember riding on the merry-go-round as a kid? Did you ever stand at the very edge of the merry-go-round and hold on tight to the railing as your friends pushed the wheel faster and faster? Maybe you remember that the faster the wheel turned, the harder it became to hold on. You might not have known it at the time, but you were creating a balance between two forces – one real and one apparent – in order to stay on that circular path.
Merry-go-rounds are a perfect example of how a force is used to keep an object moving in a circular path. Your body wanted to fly off the merry-go-round in a straight line, but your hands exerted an opposing force to keep you on. The tendency for your body to fly off the merry-go-round is called centrifugal force. It isn’t a real force, but an apparent one. The force you used with your hands to stay on the ride is real, and it is called centripetal force. Let’s learn more about it.
Centripetal force is a force on an object directed to the center of a circular path that keeps the object on the path. Its value is based on three factors: 1) the velocity of the object as it follows the circular path; 2) the object’s distance from the center of the path; and 3) the mass of the object.
Centrifugal force, on the other hand, is not a force, but a tendency for an object to leave the circular path and fly off in a straight line. Sometimes people mistakenly say ‘centrifugal force’ when they mean ‘centripetal force.’ The velocity of the object is constant and perpendicular to a line running from the object to the center of the circle; it is called tangential velocity.
An object on a circular path
Circular path diagram
In this diagram, centripetal force f is shown as a red arrow. It is constant in magnitude but keeps changing direction so that it is always pointing to the center. Also shown on the diagram is the tangential velocity, v. Finally, the constant distance of the object from the center of the circle is represented by the variable r, or radius.
How to Calculate Centripetal Force
Centripetal force is easily calculated as long as you know the mass, m, of the object; its distance, r, from the center; and the tangential velocity, v. This equation is based on the metric system; note that the centripetal force, f, is measured in Newtons. One Newton is approximately 0.225 lb.
centripetal force equation
There are some interesting things about this equation. Because the tangential velocity is squared, if you double the velocity you quadruple the centripetal force! Also, because r appears in the denominator, the magnitude of centrifugal force decreases as the object gets further away from the center. Finally, if you know the centripetal force, this equation can be rearranged to solve for velocity:
velocity equation
Let’s look at an example. Suppose in our merry-go-round scenario Erica is standing at the edge of the ride holding onto the bars. Erica weighs 70 pounds. The diameter of the merry-go-round is 3 meters. The ride is making one complete revolution every 4 seconds. What is the centripetal force Erica must exert to hold onto the ride?
The circumference of the merry-go-round is diameter multiplied by pi. This calculation gives us about 9.4 meters around the perimeter of the ride. If Erica is traveling 9.4 meters every 4 seconds, she has a tangential velocity, v, of 9.4 / 4 = 2.35 meters per second.
Next, we find the radius r by dividing the diameter by 2; r is 1.5 meters. Finally, we need to convert Erica’s body weight to a mass. On the Earth’s surface, one pound is about 0.454 kg. Thus, Erica’s body mass is 70 x 0.454 = 32 kg. Now, we are ready to use the equation:
merry-go-round example
Erica must exert a force of 118 Newtons (about 27 pounds) to stay on the ride.
Centripetal Force in Space
One of the uses of centripetal force is calculating the Earth orbit of a satellite. This has been used by scientists for decades in the space program. The idea of an Earth orbit is to keep the object moving at a fixed tangential velocity so that the force of gravity, at that distance from the Earth, is exactly equal to the centripetal force needed to keep it in orbit.
We need the formula for the Earth’s gravitational force acting on the satellite. It is:
satellite equation 1
If we set this force equal to the centripetal force equation, we get:
satellite equation 2
Notice that there is a factor of m/r on each side of the equation that can be divided out to obtain:
satellite equation 3
Solving for v, we obtain a very simple orbit design equation:
orbit equation
This equation says that regardless of the mass of the satellite, the tangential velocity needed to keep it in orbit is inversely proportional to the square root of the orbital radius.
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