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course phy 201
12/07 2:30 pm
`q001. Sketch or trace a circle whose radius we take to be A and whose center is at the origin. Make the circle big enough that you can legibly annotate the sketch you are going to make. A trace of the top of a large drinking glass, or something of comparable size, would work nicely.A point moves counterclockwise around the circle at some constant angular velocity omega, starting at the origin.
Mark the four points on the circle at which it intersects a coordinate axis (i.e., mark the two points where it intersects the x axis and the two points where it intersects the y axis).
Your four points will divide the circle into four arcs. Mark the point on each arc which is halfway between its ends. You will now have divided the circle into 8 equal arcs.
Similarly mark the midpoint of each of these 8 arcs, so that you will have marked 16 equally spaced points dividing the circle into 16 equal arcs.
Sketch a vector from the origin to each of your points.
Sketch the projection of each vector onto the y axis. Include the dotted projection lines.
Estimate the length of each projection as a percent of the radius of your circle. List your sixteen estimates. For projections which fall on the negative y axis, list your estimate as a negative number.
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going clockwise from 12:00 position; ¼, 2/3, ¾, 1, 2/3, ¼, -1/4, -2/3, -3/4, -1, -3/4, -2/3, -1/4.
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At the 12:00 position, which is the 90 degree position, the y projection of your vector would be equal to the radius. So it seems your numbers would start at 1 and decrease to 0, then to -1. The sequence
1, 2/3, ¼, -1/4, -2/3, -3/4, -1
would reasonably correspond to motion from the 12:00 to the 6:00 position.
Note also that at the 3:00 position, which is the 0 degree position, the vector is horizontal and its y projection is 0. This should also be included, so your numbers would be
1, 2/3, ¼, 0, -1/4, -2/3, -3/4, -1
Finally the positive direction is counterclockwise. That would make no difference in the numbers for the y projection, if you start from the 90 degree position.
If you started from the 0 degree position and moved counterclockwise, then your sequence
¼, 2/3, ¾, 1, 2/3, ¼, -1/4, -2/3, -3/4, -1, -3/4, -2/3, -1/4
would make good sense (though the 0's corresponding to the positions on the positive and negative x axis would also be appropriate).
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`q002. At each of your 16 marked points sketch a vector indicating the velocity of the moving point as it passes through that position. The angular velocity is constant and the moving point is always at the same distance from the origin, so its speed will always be the same. The velocity vectors will therefore all be of the same length. Since the point is moving along the circle, its velocity will at every point be tangent to the circle. You can choose any length you wish for your velocity vectors, but it is suggested that they be slightly shorter than the distance between two adjacent marked points.
Sketch a 'pretend' y axis to the right of your circle. Project each of your velocity vectors onto this y axis, and estimate the length of each projection as a percent of the length of the velocity vector. List your sixteen estimates, listing projections which point downward as negative.
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slightly confused here. Projecting onto the y-axis creates a bunch of lines going straight to the right, no up or down motion to be seen, or do I have the whole idea of projecting vectors wrong?
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You need to draw the velocity vectors around the circle, each tangent to the circle at the given point.
You would draw the y projection of each of these vectors.
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