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Uniform Circular Motion
The motion of an object moving around a circle at a constant speed can be modeled as follows.
We impose a coordinate system with the origin at the center of the circle.
The object’s position can be specified by an angle θ measured counterclockwise from the pos- tive x-axis.
Then θ = θ 0 ± ωt
where θ 0 is the angle of the initial position (i.e., the position at time t = 0), ω is the angular speed in radians per unit time, and the plus-or-minus is take to be plus if the motion is counter- clockwise, and minus if the motion is clockwise.
In xy-coordinate, the location of the object is then given by
(r cos(θ 0 ± ωt), r sin(θ 0 ± ωt))
where r is the radius of the circle.
In most problems of this sort, you will need the following:
- θ 0 (where does the thing start?)
- ω in radians (what is the angular speed?)
- the direction of motion (which way is it moving?)
- r (what is the radius of the circle?)
- the time t of interest (when are we trying to find the object’s location?)
Some of this information will be given directly to you, while other bits you’ll have to calculate.
Of particular importance is the equation
v = rω.
This ties v, r and ω together, so that if you know two of them, you can find the other. In a lot of problems v will be given; it is there so that you can find ω.
Let’s look at some examples.
Example 1 Bob starts running around from the northernmost point of a circular track. He takes 1. minutes to complete each lap of the track. The track has a radius of 200 m. He runs clockwise. Where is Bob relative to his starting point after running for one hour?
When we read the problem, we want to begin to find as many of the items in the bulleted list as possible. In the first sentence, we are told Bob starts at the northernmost point of the track: this tells us θ 0 = π/ 2.
The second sentence tells us Bob’s angular speed by telling us he makes one lap in 1.93 minutes. In one lap, he moves through 2 π radians. Hence, his angular speed is
ω =
2 π
- 93
= 3. 25554 radians per minute.
His direction we are told is clockwise, so we will go with the ”-” choice in our θ formula.
Finally, we are interested in his location after one hour, so t = 60 minutes.
Thus, after 60 minutes, we know Bob’s location is given by
θ = θ 0 − ωt =
π 2
and his coordinates after one hour are
(r cos θ, r sin θ) = (105. 1236 , 170 .1442).
Be sure your calculator is in radian mode when calculating trig values from angles in radians.
Bob’s starting point was (0, 200), so Bob’s location after one hour can be described as 105. meters east and 200-170.1442=29.8558 meters south of his starting point.
Example 2 Maria starts at the southernmost point of a circular track and run clockwise at 4 meters per second. The track has a radius of 50 m.
After running for 10 seconds, how far is she from her starting point? After 1000 seconds?
We collect the important information given in the problem:
- r = 50m
- starting position: θ 0 = −π 2 (or 32 π )
- direction of motion: clockwise (-)
