Kinematics of Uniform Circular Motion

1. Kinematics of Uniform Circular Motion Do you remember the equations of kinematics? There are analogous equations for rotational quantities. You will see them later in the course. I believe our starting point for circular motion best involves forces (dynamics). However, let’s start by considering circular motion without looking at the forces involved. Consider an object moving in a straight line. Vi

2. F V=at F V=at A force F applied parallel to the direction of motion for a time t increases the magnitude of velocity by an amount at, but does not change the direction of motion. Vi A force applied parallel to an object’s velocity vector increases the object’s speed. A force F applied perpendicular to the direction of motion for a time t changes the direction of the velocity vector. Vi In the limit t0, the length of the velocity vector does not change.

3. A force applied perpendicular to an object’s velocity vector instantaneously changes the direction of the velocity vector, but not the object’s speed. F Vf V=at Vi If the applied force is always perpendicular to the velocity vector, the object constantly changes direction, but never speeds up or slows down. In that case, the object follows a circular path.

4. Summary and consequences: A force applied parallel to an object’s velocity vector increases the object’s speed. A force applied perpendicular to an object’s velocity vector instantaneously changes the direction of the velocity vector, but not the object’s speed. If you apply a force parallel to the velocity vector you can only change an object’s speed, not its direction. If you apply a force perpendicular to an object’s velocity vector, you will change its direction of motion BUT NOT ITS SPEED!

5. An example of the latter is circular motion.  A ball tied to the end of a string and “whirled” around. A child on a merry-go-round. A car rounding a circular curve. The earth orbiting the sun (approximately). The moon orbiting the earth (approximately).

6. v An object moving in a circle with constant speed is said to undergo uniform circular motion. A ball on a string: a The instantaneous velocity is tangent to the path of motion (OK to “attach” velocity to object—this is not a free-body diagram). The instantaneous acceleration is perpendicular to the velocity vector. The ball is accelerated because its velocity constantly changes. If the motion is uniform circular, the acceleration is towards the center of the circle; i.e., the acceleration is “radial” or “centripetal.”

7. T ac The force that accelerates the ball is the tension in the string to which it is attached. The centripetal force due to the string gives rise to a centripetal (also called radial) acceleration. If the ball moves uniformly in a circle, both the force and acceleration continually change direction, so that they always point to the center of the circle. Your book shows that

8. Note: if the motion is not uniform (the speed changes or the radius of the circle changes) there will also be a tangential acceleration. We will not worry about that case here. For the problems on circular motion, you need to recall the definitions of frequency, period, and know how to use the fact that that an object moving in a circle with constant speed has velocity given by v = 2r / T.

9. Dynamics of Uniform Circular Motion Now we consider causes (forces) of circular motion. Newton’s Laws still apply. The OSE sheet contains a variation on Newton’s second law where the subscript “r” stands for “radial.” You may also use the subscript “c” (“centripetal”).

10. x Example. Suppose the ball (mass m) in the example I gave in the previous section moves with a constant speed V. What is the tension in the string. Choose an axis parallel to the acceleration vector. The direction of the y-axis is irrelevant here. T ac

11. Isn’t there an outward force, pulling the ball out? NO! What about the centrifugal force I feel when my car goes around a curve at high speed? There is no such thing as centrifugal force! But I feel a force! You feel the centripetal force of the door pushing you towards the center of the circle of the turn! Your confused brain interprets the effect of Newton’s first law as a force pushing you outward.

12. Then why do engineers (and other supposedly educated people) talk about centrifugal forces? I didn’t tell you this before, but Newton’s laws are valid only in inertial (non-accelerated) reference frames. If you wish to refer your coordinates to an axis system that is accelerated, you cannot directly apply Newton’s laws. An example of an accelerated reference frame is a child riding on a merry-go-round, from the child’s point of view.

13. If you try to apply Newton’s laws in this accelerated reference frame, it appears that there really is a “centrifugal” force “trying” to throw the child outward. Centrifugal force is a pseudo-force used to allow us to apply Newton’s laws in an accelerated reference frame. Sometimes it is much simpler to use the accelerated reference frame, so “centrifugal force” is not really a “bad” thing. Plus, it gives physicists something to nitpick. Can you think of any other common non-inertial reference frames?

14. The earth is orbiting the sun, and also revolving. It is not an inertial reference frame! Why can we use Newton’s laws? Because for “normal” problems the corrections due to rotation are small enough to be negligible. A good example is the “coriolis force,” another pseudo force. The coriolis force causes low pressure systems to rotate counterclockwise in the northern hemisphere, conterclockwise in the southern. The coriolis force does not cause your bathtub to drain with a counterclockwise rotation! It represents a “small” correction and is only observable (except for carefully-constructed experiments) for incredibly large masses of fluids.

15. Here are a couple of good links on centrifugal and coriolis forces: http://gulf.ocean.fsu.edu/~www/coriolis/coriolis.html http://www.ems.psu.edu/~fraser/Bad/BadCoriolis.html You ought to have a look at Dr. Fraser’s Bad Science page. I have caught myself in more than one mistake he talks about! http://www.ems.psu.edu/~fraser/BadScience.html

16. How a garbage disposer works:

18. Example: the moon’s nearly circular orbit about the earth has a radius of about 384000 km and a period of 27.3 days. Show that the acceleration of the earth towards the moon is 2.82x10-3 m/s2, or about 2.78x10-4 g. The mass of the moon is about 7.35x1022 kg. Show that the force that the earth exerts on the moon is 2.07x1020 newtons. What force does the moon exert on the earth? Later, if I can find the time, we will learn about universal gravitation, and see if this force agrees with the force calculated from the law of universal gravitation.