0.28 Phy1250: circular motion -- acceleration and centripetal force  (Page 3/6)

Subtracting vectors

In the earlier module titled Vector Subtraction for Blind Students , you learned how to subtract the vector named Vi from the vector named Vf using the parallelogram method.

Let the time interval get smaller and smaller

What we want to do in this module is to estimate what happens to (Vf - Vi) as the time interval, dt, gets smaller and smaller.

Given that the speed of the object remains constant, the point on the circumference of the circlethat represents the tail of Vf will move closer and closer to the horizontalaxis as the time interval gets smaller.

The angle between the vectors gets smaller and smaller

That, in turn, will cause the angle that Vf forms with the horizontal axis to move closer and closer to 90 degrees. That means that the difference in theangles that the two vectors make with the horizontal axis will grow smaller and smaller, and the angle between the two vectors will grow smaller and smaller.

The difference vector

In the earlier module, you learned that for very small angles, the difference vector is very close to being perpendicular toboth Vf and Vi. Being very close to perpendicular to both Vf and Vi means that the acceleration vector is very close to pointing directly at the center of thecircle.

Using calculus, it can be shown that in the limit, as the time interval, dt, approaches zero, the direction of the acceleration vector pointsdirectly at the center of the circle.

The magnitude of the acceleration vector

In the earlier module, you also learned that the magnitude of the difference vector approaches zero as the angle between the two velocity vectors approacheszero. However, the acceleration is equal to the magnitude of the difference vector divided by the scalar value of the time interval, dt, which is alsoapproaching zero. One very small value divided by another very small value is not necessarily a very small value. It can be shown using calculus that in thelimit, the magnitude of the acceleration vector is not zero.

The conclusion

The conclusion is that an object undergoing uniform circular motion experiences an acceleration vector that points directly at the center of thecircle and it has a non-zero magnitude.

An old-fashioned merry-go-round

That was the graphical explanation of the acceleration vector. Now I will cite a few anecdotal explanations.

A long wooden plank

When I was around ten or eleven years old, the child across the street from my house had anold-fashioned homemade merry-go-round in his back yard. This device consisted of a long wooden plank with a hole in the center. A large bolt was threaded throughthe hold and pushed vertically into a hole in the top of a post about two feet tall. The bolt formed a spindle and the plank was able to rotate around the spindle.

Two handles

Two short skinny boards were fastened to the plank about 18 inches from each end to form asort of a cross in each end. These boards extended about 9 inches on either side of the plank and were intended to be handles.