CIRCULAR MOTION 1

1. Uniform Circular Motion is when an object is moving in a circular path with a constant speed. So, if this is the definition of uniform circular motion, is an object that that undergoes UCM accelerating or not?
2. The answer is YES. Let us redefine acceleration and understand the requirements necessary for an object to be accelerating.
3. Acceleration occurs when one of the following occurs. Recall that acceleration is a vector quantity with a specified magnitude and direction. An object is accelerating if

1. its speed changes
2. the direction of the velocity vector changes
3. both its speed and direction changes.

1. If an object is moving in a circle with a constant speed, then it is accelerating since the direction of its velocity changes. This is why you might notice that you are going faster when you are rounding a curve even though you did not step on the accelerator.
2. The acceleration associated with UCM is called the radial or centripetal acceleration. Centripetal because it is "center seeking" and radial because it is directed along the radius of the circular path.
3. The figure below illustrates the relationship between the velocity vector and the centripetal acceleration in UCM. The velocity vector is always tangent to the circular path while the centripetal acceleration is directed toward the center of the circle and perpendicular to the velocity vector.




4. The magnitude of the centripetal acceleration is given by : a_R = v²/r, where v is the speed, and r is the radius of the circular path.
5. PARAMETERS ASSOCIATED WITH CIRCULAR MOTION

1. Revolution (rev) = one complete closed circular path.
2. Frequency (f) = the number of revolutions or cycles per second.
3. Period (T) = the time required for an object to complete 1 revolution.


T = 1/f or f = 1/T

4. Speed (v) = speed is the total distance divided by the total time. In this case, the total distance is the circumference of the circle (2pr) and the total time is the period (T).

1. Where there is a net acceleration, there is a net force and vice versa. There is a centripetal acceleration therefore there is a centripetal force associated with it.
2. The basic question to ask is the following: If an object is moving in circular motion, is it ever in equilibrium? The answer is NO, not along the radial direction where there will always be a centripetal acceleration.
3. Another question to ask is the following: Is the centripetal force a new kind of force? The answer is again NO. This force is just a new name for an old force. Centripetal force is must be provided or supplied by another force and the new name just tells us the direction of the force, as in toward the center of the circle.
4. For example, if I have a ball attached to a string and I whirl this ball in a horizontal circle, the force that provides the centripetal force is the Tension in the string. Similarly, if I am standing on a rotating circular platform, I experience a centripetal force supplied by the static friction keeping me in place on the platform. Another example would be the Moon rotating around the Earth. The gravitational attraction between the Earth and the Moon supplies the centripetal force for the circular motion.
5. So, how do we treat problems involving circular motion? See the simple cases drawn below.

1. An object attached to a string rotating in UCM in a horizontal circle.




2. An object attached to a string rotating in a vertical circle.

[C] NON-UNIFORM CIRCULAR MOTION

1. Non-uniform circular motion occurs when both magnitude and direction of the velocity change simultaneously.
2. We know what the change of direction gives us, a centripetal acceleration. What does the change in direction give us?
3. The change of direction gives us a tangential acceleration (a_t). This acceleration is just like the old acceleration we know from the earlier chapters: a = Dv/Dt.
4. When a particle undergoes non-uniform circular motion, it experiences both kinds of acceleration as depicted in the figure below.




5. The acceleration vector is the vector sum of both kinds of acceleration. The vector and its magnitude are given by: