Class 090928
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Fractional cycles of a pendulum
Regard the equilibrium position of a pendulum as the origin of the x axis. To the right of equilibrium x values are positive, and to the left of equilibrium x values are negative.
Suppose you release a pendulum of length 16 cm from rest, at position x = 4 cm.
Estimate its position in cm, its direction of motion (positive or negative) and its speed as a percent of its maximum speed (e.g., speed is 100 % at equilibrium, 0% at release, and somewhere between 0% and 100% at every position between) after each of the following time intervals has elapsed:
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Some Answers: After 1 complete cycle, or 4/4 of a cycle, the pendulum is back where it started, at position +4 cm and at rest.
After 1/2 cycle, or 2/4 cycle, the pendulum is at the 'extreme point' opposite its point of release and at rest. Its position is -4 cm.
After 1/4 cycle, the pendulum is halfway between the 'extreme' position of its release and the 'extreme' position opposite its point of release, so it is at the equilibrium position. It is at this position that it attains its maximum speed, so its speed at this point is 100% of the maximum. It is moving in the negative direction. It returns to this position and velocity every cycle, so the same description applies after for 5/4 cycle.
Half a cycle later, at 3/4 cycle, it is again at equilibrium, moving with 100% of maximum speed, but is now moving in the positive direction.
2/3 of a cycle is between 1/2 cycle and 3/4 cycle, so the position will be somewhere between -4 cm and equilibrium. The pendulum will be moving to the right so its velocity will be positive. 2/3 is closer to 3/4 than to 1/2, and speed builds fastest when the pendulum is further from equilibrium, but its fastest motion is still ahead. So without a mathematical model, which we develop near the end of the term, it's difficult to estimate where it is and how fast it's moving. Just about any estimate of position between -4 cm and equilibrium, and just about any speed greater than 50% of maximum, would not be an unreasonable estimate. We will find out much more later.
Similar comments apply to 7/8 cycle and .6 cycle. 7/8 lies between 3/4 and 1, so the pendulum will be between equilibrium and +4 cm, and will be moving to the right. .6 lies between 1/2 and 3/4, so the pendulum will be between -4 cm and equilibrium, and will be moving to the right.
Acceleration of Gravity
Drop a coin and release a pendulum at the same instant. Adjust the length of the pendulum so that it travels from release to equilibrium, then to the opposite extreme point and back, reaching equilibrium the second time at the same instant you hear the coin strike the floor. Measure the pendulum.
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Raw data will consist of pendulum length and distance to the floor.
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Show how to start with your raw data and reason out the acceleration of the falling coin, assuming constant acceleration:
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Pendulum length will give you the period.
It should be clear that the pendulum oscillated through 3/4 of a cycle.
So you can determine the time of fall. You know the initial velocity was zero, so you know v0, `dt and `ds. You can reason out the acceleration using the definitions, or use the equations of motion. You should be able to do this either way.
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There are two delays between the events you are observing and your perceptions:
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Introduction to Projectile motion
Time a ball down a ramp, and measure how far it travels in the horizontal direction.
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Your raw data would include pendulum length and number of half-cycles or cycles (you should specify which). You would also specify the distance down the ramp and the horizontal displacement after leaving the edge of the ramp, as well as the distance to the floor.
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To keep things straight, let's use the following notation in the rest of this analysis:
Note that your raw data include `ds_ramp, `ds_x_projectile and `ds_y_projectile. You also have the number of cycles or half-cycles down the ramp, which with your pendulum length allows you to directly calculate `dt_ramp.
You do not have observations of vf_ramp or `dt_projectile, which must be calculated from the observed information using direct reasoning and/or the equations of motion.
Answer the following questions:
According to the time `dt_ramp required to travel down the ramp and its length `ds_ramp, what are the average and final velocities on the ramp, assuming uniform acceleration?
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The average velocity would be obtained by dividing `ds_ramp, the displacement down the ramp, by `dt_ramp, the time spent on the ramp, as determined by your pendulum. Since the initial velocity is zero and acceleration is uniform, you can easily determine the final velocity.
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Moving at vf_ramp, how long would it take the ball to travel through displacement `ds_x_projectile?
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You would divide the observed horizontal displacement of the projectile, `ds_x_projectile, by its horizontal velocity, which we assume here to be the same as vf_ramp.
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Accelerating at 1000 cm/s^2, how long would it take the ball to fall from rest through displacement `ds_y_projectile?
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The ball falls from rest through a vertical displacement you measured, at the given acceleration. You know v0, `ds and a. You can use the equations of motion to find everything else.
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In the time interval you just calculated, how far would the ball travel if moving at velocity v_f_ramp?
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You would multiply the time interval, as calculated in the preceding question, by the horizontal velocity, which we assume to be constant and equal to vf_ramp.
You would not use the time required to roll down the ramp. That time interval and the time the ball spends falling to the floor have nothing to do with one another.&&&&
Accelerating at the rate you calculated in the preceding exercise, how long would it take the ball to fall from rest through displacement equal to `ds_y_projectile?
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This question is pretty ambiguous; it doesn't make it clear what 'the preceding exercise' is. You can ignore this one for the moment.
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Ball up and down ramp
'Poke' a ball (perhaps using your pencil as a 'cue stick') so that it travels
partway up a ramp then
back. Observe the clock time and position at three events: the end of the
'poke', when the ball comes
to rest for an instant before rolling back down, and its return to its original
position.
Report your raw data:
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Your raw data consist of the number of half-cycles required to come to rest, then the number required to roll back to the starting point, as well as the distance from the starting point to the point of rest.
You cannot assume that the two time intervals are the same, that it requires half the total time to go up the ramp and the other half to come back down. For an ideal situation this would in fact be so, but this is the real world and doesn't necessarily behave in the ideal manner.
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Declare your positive direction.
Determine the initial velocity and acceleration of the ball
for the interval between the first and second event.
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For this interval you know the time interval. The final velocity is zero, and you measured the displacement. You can reason this out or use the equations.
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Determine the final velocity and acceleration of the ball for the interval between the second and third event.
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Here you know the initial velocity (zero), the displacement and the time interval. You can reason this out or use the equations.
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Do you think the acceleration of the ball is greater between the first and second event, or between between the second and third event? Or do you think it is the same for both? Give reasons for your answer.
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Note that if you declare your positive direction and do your calculations correctly, the acceleration will have the same sign for both intervals, and this sign will agree with the direction down the ramp.
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Are your data accurate enough to determine on which interval the acceleration is greater? If so, on which interval do you determine it is greater? If not, how accurate do you think your data would need to be to decide this question?
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Homework:
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