Uniform Circular Motion

The centripetal force in this case is provided entirely by the tension in the string. If the maximum value of the tension is 50 N, and the radius is set at 10 m we only need to plug these two values into the equation for centripetal force:

thus

Since F _c varies with v ² , an increase in velocity by a factor of two must be accompanied by an increase in centripetal force by a factor of four.

The acceleration felt by any object in uniform circular motion is given by a = . We are given the radius but must find the velocity of the satellite. We know that in one day, or 86400 seconds, the satellite travels around the earth once. Thus:

thus

Again, we use the equation F _c = . Rearranging, we find that r = . Plugging in the maximum value for the lift of the plane, we find that

During the entire trip, the rider experiences two different forces: the normal force from the track, and the gravitational force. At the top of the loop, both these forces point down, or towards the center of the loop. Thus the combination of these forces provides the centripetal force at that point. At the minimum speed of the motorcycle, however, he experiences no normal force. One can see this by envisioning that if the rider had gone any slower, he would have fallen off the track. Thus, at the minimum speed, all the centripetal force is provided by gravity. Plugging into our equation for centripetal force, we see that

Rearranging the equation,

Thus the rider must be traveling at least 9.9 m/s to make it around the loop.