0.28 Phy1250: circular motion -- acceleration and centripetal force  (Page 5/6)

Water in a bucket

If you don't mind taking a chance on getting wet, fill a small bucket about half full of water. Then swing the bucket rapidly in a circle in a vertical plane.

If you can swing the bucket fast enough, the water will stay in the bucket even when the bucket is upside down. Why is that? The water molecules want tomove in a straight line. However, the inside surface of the bucket exerts a centripetal force on the water molecules causing them to accelerate toward thecenter of the circle. The centripetal force increases with the speed of the bucket. As long as the centripetal force is greater than the weight of thewater, the water won't fall out of the bucket onto your head. However, if you allow the speed of the bucket to decrease, you will reach the point wheregravity will overcome, and you will probably get wet.

The moon and the Earth

Don't ask me how it got started in the first place, but somehow the moon got started circling the Earth.

The speed of the moon and the radius of its orbit is just exactly right so that the gravitational force that the Earth exerts on the moon causes the moonto accelerate towards the Earth. The amount of acceleration toward the earth, when combined with the speed of the moon, causes the moon to move in a uniformcircular orbit around the Earth instead of either flying off into space or crashing into the Earth.

The work-energy explanation

It is probably time to start explaining this phenomena in a more technical and less anecdotal manner.

The application of a centripetal force for uniform circular motion causes the direction of the object to be changed without changing its speed. Let's see if we can explain this froma work-energy viewpoint.

Work

Recall from an earlier module that work is a force acting upon an object to cause a displacement . Also recall that the amount of work done on an object, expressed in Joules, is given by

Work = F * D * cosine(theta)

where

• F represents the force that causes the displacement expressed in Newtons.
• D represents the amount of the displacement expressed in meters.
• theta is the angle between the direction of the displacement and the line of action of the force.

Centripetal force points toward center of circle

As we showed (or claimed to show) earlier, the centripetal force for an object in uniform circular motion always points in the direction of the center of the circle. At the sametime, the velocity vector, which represents the direction of the displacement is tangential to the circle. Therefore, the angle between the centripetal force andthe direction of displacement is 90 degrees. This means that the centripetal force does no work on the object, because the cosine of 90 degrees is zero.

No work is done

When no work is done upon an object by external forces, the total mechanical energy, consisting of potential energy plus kinetic energy, of the objectremains constant. If an object is moving in a circle in a plane that is parallel to the surface of the Earth, the effect of gravity may pull the entire planetoward the surface of the Earth, but it won't effect the circular motion unequally with respect to the position of the object in the circle.