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course phy 201
11/19 2:00 am
Many of the questions below can be answered quickly. Some will require more work and more thought. There are about 30 questions here, more or less, spread over five problems (`q001 - `q005). I suggest you work within a limit of 5 minutes per question on your first pass through this document and submit what you have at that point. Partial answers, and/or questions, are welcome. I'll then be able to give you feedback, hints, etc. to help you move ahead.`q001. Report your data from the experiments conducted today.
Trial one: 16 seconds, 1395 degrees (24.34 radians)
Trial two: 17 seconds, 1350 degrees ( 23.55 radians)
Trial three: 13 seconds, 1350 degrees (23.55 radians)
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For the experiment where the domino slipped off the strap if it was going too fast, what was the maximum angular velocity of the system for which the domino did not slip off?
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1.81 radians/second;
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How far was the domino from the axis of rotation?
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10 cm
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How fast was the domino therefore moving?
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ds = r* theta; 10 cm * 23.55 radians = 235.5 cm; 235.5 cm/13 seconds = 18.12 cm/s;
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The centripetal acceleration of an object moving with speed v around a circle of radius r is
a_centripetal = v^2 / r.
The centripetal acceleration of the domino is the acceleration toward the center required to keep it moving in a circular path. What was the maximum centripetal acceleration, among your trials, for which the domino did not slip off?
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centripetal acceleration = v^2/r; 1.81^2/10; 0.327 cm/s^2;
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I think 1.81 rad/sec is your average, not your maximum angular velocity. The maximum, since the strap was slowing, would have been the initial angular velocity.
I therefore expect your maximum angular velocity was double what you used, which would quadruple the present result.
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What was the coefficient of friction between domino and strap, based on the slope required for the domino to begin sliding?
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2.5/30.4; based off data, roughly 8.22%
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again this should probably be quadrupled, per my previous note
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About 32% would be consistent with results obtained by other groups, and with what I would expect for this situation.
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Assuming that the ball in the experiment required .42 seconds to fall to the floor, what speeds do you conclude for each trial?
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only two values. The landing spot was the essentially the same for all trials without the magnet, and they all clustered within a reasonable distance (less than a centimeter) with the magnet.
Trial one (no magnet): 11.5 cm directly outward. Velocity: 27.38 cm/s
Trial two (with magnet): 12.5 cm, slightly to the right. Velocity: 29.76 cm/s
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Do you conclude that the magnet sped the ball up, slowed it down, or that it had no significant effect?
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sped the ball up
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For each set of trials, how much speed did the magnet induce in the direction perpendicular to the original direction?
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sideways distance: 2.5 cm; horizontal velocity: 5.95 cm/s;
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The magnet is 5 cm long. Assuming that the force exerted by the magnet has a significant effect for only this distance, for what time interval was the ball influenced by the magnet?
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12.5 cm/s; 0.4 seconds;
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You calculated horizontal velocities close to 30 cm/s.
When the ball passed the magnet is was very near the end of the ramp and would not have had time to gain much more speed, so the velocity as it passed the magnet would have been close to 30 cm/s.
This would reduce the time near the magnet to about .17 second.
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By how much did the momentum of the ball change due to the effect of the magnet, based on the velocity it attained perpendicular to its original line of motion? Assume the ball's mass to be 20 grams.
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20 grams * 1 cm = 20 gram cm;
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Momentum is mass * velocity.
1 cm isn't a velocity.
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What do you conclude was the average force exerted on the ball by the magnet?
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momentum = impulse; momentum: 20 gram cm; impulse = force * dt; dt = 0.4 seconds; 20/0.4 = 50 gram cm^2/s^2
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20 has units, as does 0.4.
The units you have given these quantities don't work out to gram cm^2 / s^2, nor do the correct units.
However your general idea, which would be to divide the momentum change by the time interval, is correct.
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`q002. The magnet and domino on the strap were located 10.5 cm and 17 cm from the axis of rotation. On one trial the system rotated through 210 degrees in 6 seconds, ending up at rest.
What were the average and initial angular velocities of the system, in degrees / second, if we assume uniform angular acceleration?
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vAve: 35 degrees/second; v0: 70 degrees/second;
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What is the circumference of each of the circles around which the magnet and domino traveled?
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magnet: roughly 66 cm;
domino: 106.7 cm
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How far did each actually travel during the 210 degree rotation?
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magnet: roughly 38.5 cm;
domino: roughly 62.2 cm;
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What therefore was the average speed of each around the arc, in units of distance / time?
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magnet: 6.42 cm/s
domino: 10.37 cm/s
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What was the initial speed of each?
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magnet: 12.84 cm/s
domino: 20.74 cm/s
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`q003. A radian is the angle subtended by the arc of a circle whose arc length is equal to the radius of that circle.
What are the lengths of the arcs of a circle of radius 15 cm corresponding to angles of 2 radians, 1/2 radian and 6 radians?
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13 cm, 7.5 cm, 90 cm
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What are the angles subtended on the same circle by arcs of 45 cm, 5 cm and 25 cm?
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3 radians, 0.33 radians, 1.67 radians
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I'll bet you meant that 13 to be 30. Everything else is fine.
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What is the angle subtended by an arc consisting of the entire circumference of the circle?
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2*pi
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The angle subtended by an arc consisting of the entire circumference of any circle is 360 degrees. That angle is also 2 pi radians.
How many degrees are therefore contained in a radian?
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2 pi radians = 360 degrees; pi radians = 180 degrees; 180 degrees/pi = radian; roughly 57.3 degrees = radian
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How many radians are there in a degree?
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roughly 0.017 radians
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Returning to the preceding problem, where the strap rotated through 210 degrees in 6 seconds, through how many radians did the strap rotate?
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roughly 3.57 radians
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Based on your result for the number of radians, what were the average and initial angular velocities of the strap?
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average: 0.595 radians/sec
initial: 1.19 radians/sec
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Your answers to the preceding question would be in radians / second. If the initial angular velocity was maintained, through how many radians would the system rotate in one second?
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1.19 radians
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Recall that one radian of angle corresponds to an arc distance equal to the radius. What arc distance would therefore correspond to one second's worth of rotation for the domino, were the initial angular velocity to be maintained?
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17 cm * 1.19 = 20.23 cm
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What arc distance would correspond to one second's worth of rotation for the magnet, were the initial angular velocity to be maintained?
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10.5 * 1.19 = 12.495 cm
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How fast are the domino and the magnet therefore moving at the initial instant?
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domino: 20.23 cm/s
magnet: 12.495 cm/s
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`q004. Assume that the center of the strap is halfway between the domino and the magnet. How far then is the center of the strap from the axis of rotation?
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center of strap: 13.5 cm
axis of rotation: 3.5 cm away from center;
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If the strap has mass 70 grams then what is its torque about the axis of rotation? The torque is equal to the weight of the strap multiplied by the distance of its center from the axis of rotation.
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70 * 3.5 = 245 gram cm
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If the domino has mass 17 grams then what is its torque about the axis of rotation?
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17 grams * 17 cm = 289 gram cm
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Are the domino and the center of strap both on the same side of the axis of rotation, or on opposite sides? Do their torques therefore reinforce or work counter to one another?
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same sides; The axis of rotation is closer to the magnet than to the center;
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What therefore is the total torque exerted by strap and domino?
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534 gram cm
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The torque produced by the magnet is equal and opposite to the sum of the other torques. What therefore is the magnitude of that torque, and what do you conclude is the mass of the magnet?
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torque: 534 gram cm; distance: 10.5 cm; mass of magnet: 53.4 grams
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Very good overall. Check my notes for a couple of anomalies and let me know if you have questions.
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