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CSET Science: Study Guide & Test Prep26 chapters | 314 lessons

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Lesson Transcript

Instructor:
*Scott van Tonningen*

Scott has a Ph.D. in electrical engineering and has taught a variety of college-level engineering, math and science courses.

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.

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**.

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.

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.

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:

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:

Erica must exert a force of 118 Newtons (about 27 pounds) to stay on the ride.

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:

If we set this force equal to the centripetal force equation, we get:

Notice that there is a factor of *m*/*r* on each side of the equation that can be divided out to obtain:

Solving for *v*, we obtain a very simple orbit design 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.

Let's do an example based on one of the 24 satellites in the Global Positioning System (GPS) constellation. These satellites orbit at an altitude of 20,000 km above the Earth. Add to this the radius of the Earth, which is about 6378 km, and you get an orbital radius of 26,378 km. Let's use the orbit design equation to find the required tangential velocity:

*v* = (1.996 x 10^7) / (?(26,378 x 10^3)) = 3,872 meters/sec

At this velocity and orbital radius, the GPS satellite will orbit the Earth exactly twice in one 24 hour period.

Here is an image of a centrifuge developed for NASA that can create 20 times the force of gravity on a subject riding in the machine. The centrifuge rotates about a central axis very rapidly to create an apparent centrifugal force. The centripetal force is equal and opposite and is provided by the structure of the compartment that the subject is sitting in. The centrifuge is used to give subjects experience in the tremendous G forces associated with space travel.

If the NASA centrifuge has a radius of 29 feet and turns at a rate of 45 RPM (revolutions per minute), how much force would be exerted on a 170-pound human?

29 feet is 8.84 meters. The circumference of the circular path is 2 x 8.84 x pi, which is 55.55 meters. We can use this equation to find the tangential velocity:

Using the fact that 1 pound is 0.454 kg, the 170-pound human would have a mass of 77.18 kg. Now, we can apply the centripetal force equation:

One Newton is about 0.225 pounds, so this force is the equivalent of 3,409 pounds, or about 20 times the weight of the human.

**Centripetal force** is the force on an object on a circular path that keeps the object moving on the path. It is always directed towards the center and its magnitude is constant, based on the mass of the object, its tangential velocity, and the distance of the object (radius) from the center of the circular path.

Centripetal force counteracts **centrifugal force**, which is not a real force but is the tendency for objects on a circular path to leave the path in a straight line. Finally, **tangential velocity** is the constant velocity of the object that is perpendicular to the line running from the object to the center of the path.

Centripetal force is measured in Newtons and is calculated as the mass (in kg), multiplied by tangential velocity (in meters per second) squared, divided by the radius (in meters). This means that if tangential velocity doubles, the force will quadruple. It also means that centripetal force decreases as the radius gets larger.

There are many examples of centripetal force such as merry-go-rounds, Earth orbits, and centrifuges. In Earth orbits, the force of gravity is equal to the centripetal force and keeps the space object moving in the orbit. This fact leads to a very simple orbital design equation.

Centripetal force:

- Is the force on an object on a circular path that keeps the object on its path
- Counteracts centrifugal force
- Is equal to the mass (kg) x tangential velocity (meters per second) / radius (meters)

This lesson on centripetal force could prepare you to:

- Understand the meaning of centripetal force
- Compare centripetal and centrifugal force
- Calculate centripetal force

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CSET Science: Study Guide & Test Prep26 chapters | 314 lessons

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