course phys 241
This looks good.
Physics 241
Acceleration of object moving in circle
For this experiment, I tied a washer to a piece of dental floss 1.2 meters long. I made marks on the string at .2 m, .4 m, .6 m, .8 m, and 1 m. I rotated the washer with the string at the various lengths with each rotation being 10 cm from the ground. The median results of the releases are as follows.
String length Distance from ground Distance traveled Velocity
20 cm 50 cm 34 cm 136 cm/sec
40 cm 70 cm 90 cm 281.25 cm/sec
60 cm 90 cm 180 cm 473.68 cm/sec
80 cm 110 cm 220 cm 511.63 cm/sec
100 cm 130 cm 255 cm 542.55 cm/sec
As the graph shows, the velocity of the projectile at each string length is increasing at a decreasing rate. For the next part of the experiment, I had to linearize the graph. The table shows the results of squaring the velocity and finding the square root of the velocity.
String length (radius) Velocity Velocity^2 Velocity^.5
20 cm 136 cm/sec 18496 cm^2/s^2 11.66 cm^2/s^2
40 cm 281.25 cm/sec 79101.56 cm^2/s^2 16.77 cm^2/s^2
60 cm 473.68 cm/sec 224372.74 cm^2/s^2 21.76 cm^2/s^2
80 cm 511.63 cm/sec 261765.26 cm^2/s^2 22.62 cm^2/s^2
100 cm 542.55 cm/sec 294360.50 cm^2/s^2 23.29 cm^2/s^2
As you can see from the two graphs, both squaring the velocity (left) and finding the square root of the velocity (right) do a decent job of linearizing the graph. For this experiment, I will use the square root of the velocity as the power function. The graph shows the relationship of radius vs Velocity^.5. In this graph, the slope is .15 cm^2/s^2 for every cm, which equals k. this means that 1/k is 6.67. The results of my experiment seem to validate the theory that the centripetal acceleration at the top of the rotation is 9.8 m/s^2.
There are of course levels of error associated with every experiment, and this one is no different. I would speculate that I could have been as much as 5 degrees off from vertical release during any one of the launches. This could cause the projectile to either hit the ground prematurely or later than it should, depending on the position at time of release. I believe that the string went slack at the approximate vertical center of the rotation. If string’s slack was off, it would also cause more or less acceleration depending on if I released the projectile before or after this slack. I had an assistant help me for this experiment, so I am fairly confident in the horizontal spotting of the projectile. An error in this spot would cause the velocity calculations to be incorrect. The area where there is the most concern for error is the vertical height of my arm during the rotation of the projectile. This would cause the distance and time elements to be erroneous, and could have serious affects on the results of the experiment. I feel that the most error that could have occurred is less than 5 cm, which does not affect the results to a large extent.
As I stated earlier, the greatest cause for error in this experiment for me is the vertical height of my arm. This could be as much as 5 cm. The next most crucial possible error is the point of release for the object. After that, the next error that had an affect is the exact location where there was slack on the line. The most insignificant error is the actual horizontal spotting of the washer.
The possible variations in the horizontal spot of the object is less than 5 cm from center. This could cause the velocity of the object to fluctuate by as much as ten cm in the measured time. This however, is unlikely. The variation would cause a change in the v^p ratio by the difference in velocities. The maximum values and minimum values of velocity are:
Maximum slope Minimum slope
5.33 5.10
A more realistic setup for this experiment would be to set up contact paper on the floor. Set up a ruler at a room corner, use it as a stable area to turn your object while still keeping your hand at the appropriate height. This would eliminate error in the height of my hand. Release the object, and watch as it lands on the contact paper. This would eliminate error in the horizontal distance of the projectile. Watch the rotation, and release the string just as the washer passes the corner of the wall. This would ensure that the object is released at the absolute top of the rotation. The only way to ensure that the object is released when there is slack in the line is to set up some sort of scale so that the pressure could be measured, and I wait until the pressure drops at the top of the rotation. This would reduce the error at top of rotation.