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course phy201
`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.
-1/8,-1/6,-1/10,-1/4,1/8,1/6,1/10,1/4
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If you start from the point on the positive x axis and go counterclockwise around the circle the y coordinates of the vector from the origin to the reference point would be increasing and positive up to the 90 degree position, then they would be positive but decreasing.
At the 0 degree position the position vector would lie along the x axos, so the y component of the position vector would be zero. At the 90 degree position the y component of the position vector would be equal to the radius. The numbers at the 3 in between points would be increasing, and all numbers would be between 0 and 1. Between the 90 degree and 180 degree positions the vectors would decrease to 0.
Between 90 degrees and 360 degrees the y components of the position vector would decrease through negative values to -1, the increase through negative values to 0.
Your values show both positive and negative values, indicating that you probably have a good diagram, but it's not clear in what direction you are going around the circle or from what point you are starting. The values 1/8, 1/6, 1/10, 1/4 increase, then decrease, then increase so they don't appear to be in an order that corresponds to the circle.
Show me your picture for this problem in Monday's class and let's see if we can resolve the order of the points.
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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.
1/16,1/10,1/8,1/4,-1/16,-1/10,-1/8,-1/4
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These values are very plausible, assuming that your velocity vectors are each represented by an arrow whose length is 1/4 of the radius of the circle.
However between two consecutive points the y component would not suddenly jump from 1/4 to -1/16.
Starting at the 0 degree position a more plausible order would be 1/4, 1/8, 1/10, 1/16, -1/16, -1/10, -1/8, -1/4.
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`q003. Sketch another copy of your circle and the 16 points (you might want to just trace your original), and for each of the 16 marked points sketch a vector representing the centripetal acceleration at that point. The centripetal acceleration will, as you should know, always point toward the center of the circle. Make your vectors about as long as the velocity vectors you used on your previous sketch (acceleration, velocity and distance are completely different quantities so the scales of the three are not related; only the distance scale is dictated by your sketch so the velocity and acceleration vectors can be sketched with any length you choose). The velocity is constant so the centripetal acceleration is constant, so all your acceleration vectors will have the same length.
Sketch the projection of each acceleration vector onto the y axis, and estimate each projection as a percent of the length of the acceleration vector. List your 16 estimates, listing projections which point downward as negative.
-1/10,-1/8,-1/6,-1/4,1/10,1/8,1/6,1/4
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From the 0 degree to the 90 degree position your order
-1/10,-1/8,-1/6,-1/4
makes sense. At the 0 degree position the y component would be 0 so the complete list might be
0, -1/10,-1/8,-1/6,-1/4.
Then the y components might decrease back to 0 so that the values from the 0 degree to the 180 degree position would read
0, -1/10,-1/8,-1/6,-1/4, -1/6, -1/8, -1/10, 0.
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`q004. If an ideal rubber band chain has a force vs. length graph with slope k, then an object of mass m suspended from the chain will oscillate naturally is a way represented by a 'reference point' moving around a circle of appropriate radius, in the manner of the circles you have just sketched. The angular velocity of the reference point will be
omega = sqrt(k / m).
A typical rubber band chain of the types we have used in class might have a force vs. length graph with a slope of 20 Newtons / meter.
If we suspend a bag containing ten 15-gram dominoes from the chain, what will be the angular velocity of the reference point?
Omega=11.54 m/s
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You don't indicate how you got this. I get a result closer to 3.5 rad / sec.
Note that m/s is not an appropriate unit for an angular velocity, and the correct calculation will not yield m/s.
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How long will it take the reference point to go all the way around the circle?
11.54m/s
V=ds/dt
11.54m/s x 2pi
72.5 s
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11.54 m/s * 2 pi = 72.5 m/s, not 72.5 second.
You don't multiply how fast by how far to figure out how long something takes. At 11.54 rad / sec, how long would it take to go through 2 pi radians?
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How many times will the reference point go all the way around the circle in one minute?
11.54m/s=ds/60s
Ds=695.4 m
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At 11.54 m/s, this is how far you would travel in 60 seconds.
However it's not the number of times you go around the reference circle in 60 seconds.
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How would your answers to these questions differ for a bag containing only half as many dominoes?
Omega would be different and distance traveled in 1 minute
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University Physics Students (and ambitions General College Physics students; this question doesn't require calculus):
What would be the answers to the above questions cut the rubber chain in half and suspended the ten dominoes from one of the halves?
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What would be your answers if you held one end of each half of the cut chain between your fingers, and suspended the ten dominoes from the other end?
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