Centripetal Force

To give an object moving in a circle the centripetal acceleration needed to keep moving, it needs a force applied to it. Any force that causes an object to move in a circle is a centripetal force. Gravity, tension, friction, and other forces can all act as centripetal forces; all of these forces can act to pull or push an object into a circle.

Because F = ma, the centripetal force F_c is just ma_c. Here’s the equation for centripetal force:

You can also calculate the centripetal acceleration a_c using the angular velocity:

This means that you can also calculate the centripetal force with the following formula:

Sample question

1. The moon goes around Earth about every 27.3 days with a distance from Earth of 3.85 x 10⁸ m. If the moon’s mass is 7.35 x 10²² kg, what is the centripetal force that Earth’s gravity exerts on it as it orbits Earth?

   The correct answer is 2.0 x 10²⁰ N.

   1. Start with this equation:

      

   2. Find the speed of the moon. It goes 2ðr in 27.3 days, so convert 27.3 to seconds:

      

   3. Therefore, the speed of the moon is

      

   4. Plug in the numbers:

      

Practice questions

1. You’re exerting a force on a string to keep a ball on a string going in a circle. If the ball has a mass of 0.10 kg and the angular velocity of the ball is 8.0 radians/s at a distance of 2.0 m, what is the centripetal force you need to apply to keep the ball going in a circle?

2. You have a 1.0-kg toy plane on the end of a 10-m wire, and it’s going around at 6.0 radians/s. What is the force you have to apply to the wire to keep the plane going in a circle?

Following are answers to the practice questions:

1. 13 N

   1. Use the equation for a centripetal force:

      

   2. Plug in the numbers:

      

2. 360 N

   1. Use this equation for a centripetal force:

      

   2. Plug in the numbers: