**Lab 114: Uniform Circular Motion**

__Introduction__

Under conditions of centripetal motion, a body with mass m traveling a circular path of radius r around a point will have a linear velocity tangent to the circle. According to Newton’s First Law of Motion, this velocity will be maintained unless the object is acted upon by external forces. A constant force acting normal to the path would produce a constantly changing direction of leaving the speed unaffected.

Supposing a body is traveling in a circular path with constant speed, acceleration, called centripetal acceleration in this special case, will be directed towards the center of the circular path. According to Newton’s Second Law, F = ma, and force and acceleration act in the same direction, so obtaining the centripetal acceleration means centripetal force can be calculated.

In centripetal motion, the acceleration is calculated with the equation a = v^{2}/r. other equations provided for this lab include:

v = r ω , v = r (2πn), where v is linear velocity, ω is angular velocity, and r is the radius of motion.

A_{c} = r ω^{2}, A = 4π^{2}n^{2}r, where A_{c} is centripetal acceleration, r is the radius of motion, and ω represents angular velocity.

F_{c }= 4 π^{2}mn^{2}r where F_{c} is the centripetal force, m is the mass of the rotating body, n is the number of rotations per second, and r is the radius.

In accordance with Hook’s Law, the external force necessary to produce a deformation d in an elastic body is given by the equation: F = kd, where the force constant k is defined as the force necessary to cause unit deflection.

In this experiment, we will verify the equation and relationship F_{c }= 4 π^{2}mn^{2}r by comparing the computed values of the centripetal force using this equation with the static force required to displace the mass to the same radial position based on the equation F = kd. In doing so, the equation for centripetal acceleration: A_{c} = r ω^{2 }will also be confirmed.

__Experimental Procedure__- The base of the apparatus was adjusted to a horizontal position.
- The mass of the revolving mass was recorded as 449 grams.
- The radius of rotation was set by adjusting the radial indicator rod. Using a ruler, the length of the radius was recorded.
- The position of the crossarm was slightly modified so that the revolving mass hanged freely over the indicator when the spring was detached. The position of the counterweight was also revised so that the crossarm was horizontal for each trial.
- The static force required to displace the mass to the same radial position was determined by passing a string over the pulley with one end of the string attached to the rotating mass and the other end of the string hooked with a weight hanger with weights attached.
- With the revolving mass attached to the spring, the system was rotated at a constant speed, taking special care that the mass passed over the indicator.
- 50 revolutions of motion for each trial was timed with a stopwatch.
- The radius of motion was changed for 2 more trials. The above steps were repeated.
- The centripetal force and acceleration were calculated with given equations.
- The calculated force and measured force were compared for difference.
- Data was recorded in the data table.
__Results (Data and Conclusion)__

__4.1 Experimental Data__

Trial # |
Mass(kg) |
Radius(m) |
Time for 50 Revolutions (s) |
Avg.time for 50 Revolutions(s) |
RPS (n)(rotations per second) |
Force Computed (N) |
Force Measured (N) |
% Diff. |

1 |
0.449 |
0.154 |
47.93 |
48.82 |
0.98 |
2.62 |
5.40 |
106% |

53.50 |
||||||||

45.02 |
||||||||

2 |
0.449 |
0.194 |
51.34 |
55.28 |
1.11 |
4.24 |
4.91 |
15.8% |

53.52 |
||||||||

60.98 |
||||||||

3 |
0.449 |
0.174 |
43.00 |
41.9 |
0.84 |
2.18 |
5.40 |
148% |

42.77 |
||||||||

39.93 |

__4.2 Calculations__

For RPS, or rotations per second, each trial’s RPS was arrived at by dividing the avg. time for 50 revolutions by 50. Force computed was calculated with the given equation F_{c }= 4 π^{2}mn^{2}r. Sample calculations for trial 1 are as followed:

m = 0.449 kg

n = 0.98 RPS

r = 0.154 m

F_{c }= 4 π^{2}mn^{2}r

F_{c }= 4 π^{2}(0.449 kg)( 0.98 RPS)^{2}(0.154 m)

F_{c }= 2.62 N

Force measured was simply derived by taking the weight of the counterweight for each trial and multiplying by 9.81 m/s^{2}.

Percent difference calculations will be included in the next section, on error.

__4.3 Error Analysis__

There was significant error in this experiment. Rotating the apparatus at constant speed was not achieved evenly between the different trials, as seen in the data. In addition to that, the force computed in many trials did not match very closely to force measured. Likely the instrument used was not very sensitive to changes in weight so the measurements were off.

Sample error calculations: (for trial 1)

Theoretical/True Value:

Force Computed: 2.62 N

Experimental Value:

Force Measured: 5.40 N

__Trial 1 error__

E_{abs} = observed value – true value

E_{abs} = 5.40 N – 2.62 N = 2.78 N

5.40 N ± 2.78 N

E_{rel} = E_{abs }/true value

E_{rel} = 2.78 /2.62 N

E_{rel} = 106%

__Discussion__

The computed force values were not close at all to the experimental forces. In most if not all trials, the apparatus was spinning at a much faster rate than recommended for optimal results. Trial 2 was the only decent result, with a 15.8% error, the other 2 trials averaging around 120% error. Since we were supposed to simulate a condition of constant speed, the trials should have been run a little slower. Even small changes in RPS would have had an impact on the final computed force because the force equation requires that RPS is squared for calculations. Had we performed the experiment more carefully, with a slower speed that would be more likely to remain constant throughout the entire time interval, the results probably would have been a lot more accurate. The error was significant, and to prevent this same problem next time and improve upon these results, I would suggest paying more attention to the revolution speed and making sure the same speed is sustainable for 50 revolutions before starting the stopwatch.

Besides that, sources of error could have included errors of reading the time, length, or mass measurements from the various equipment used, error in manually recording and measuring data, failing to perform an adequate number of trials, error in setting up the experiment, incorrectly adding up weights, failing to adjust the base of the apparatus for the top bar to horizontal for each trial, forgetting to attach the weight to the spring before starting a trial, among other things. Systematic error could have occurred because supposing the spinning apparatus was tilted or there was some other inconsistency not accounted for, results could have been slightly skewed up or down. The stopwatch or other measuring equipment could be malfunctioning for varying results. Random errors could have occurred as well. Different people performing different trials meant that characteristic ways of spinning the apparatus could have affected the results to be a little lower or higher. It could have also occurred that different people started counting the revolutions from different vantage points, so the time recorded would vary slightly.

The string connecting the mass to the vertical shaft or center bar of the spinning apparatus provides the centripetal force on the body of mass m when it undergoes circular motion. There is always a force towards the rod because the rod restricts the motion of the mass so that instead of heading straight in one direction, the mass keeps on turning because of the tension force on the string connecting the rod and the mass.

Equation 5 or the equation A_{c} = r ω^{2}, A_{c} = 4π^{2}n^{2}r for centripetal acceleration would have been verified had the equation results been more accurate. Many trials were very off; however, trial 2 was close enough to the experimental findings that the relationship in this equation as calculated can be proven correct. The same goes for the equation for centripetal force, F_{c }= 4 π^{2}mn^{2}r.

It is not possible for there to be an acceleration when velocity is constant. If velocity is constant, then since velocity is a vector quantity, both quantity of velocity and direction of velocity must remain constant. If speed is constant, the only other way for an acceleration to exist is if there is a changing direction of motion, which is not true in this situation. If speed is constant, however, that creates no assumption about changing directions, which means that an acceleration and its existence is possible.

The radius of motion at the North Pole is much smaller than at the equator of the earth. Since the equation for centripetal acceleration is A_{c} = r ω^{2}, where angular velocity is the same for any point on the same axis, as applicable to both situations, the situation with the higher radius is going to have a greater acceleration. Knowing that, the centripetal acceleration is greater at the equator of the earth.

Ball B’s string is likely to break first. 1 revolution takes the same amount of time at any radius around the axis. But since this is true, two objects at varying distances from the axis of revolution must cover differing distances during the same amount of time. Knowing that, object B must cover a larger distance, and as a result, must have a greater velocity than object A. A faster object is more liable to break off a string it revolves on.

Centripetal acceleration is a special kind of acceleration where a force is pointed perpendicular to the direction of motion. This force changes the direction of the velocity, but doesn’t change the velocity’s strength. Under this special condition, the moon can constantly accelerate towards the earth, but never fall towards earth.

__Conclusion__

As this lab is concluded, I now understand the concept of uniform circular motion. I know how to study the motion of a body traveling in a circular path at constant speed. In addition to that, I also know how to apply equations for centripetal acceleration and force so that they hold true under experimental conditions. This lab may not have been performed as ideally as liked, but I understand the possible error factors involved and were the experiment performed again, I would make a greater effort a second time around to spin the apparatus at a sustainable constant speed for a more accurate time measurement.

For this lab, we performed 3 trials in observing the relationships found unique to centripetal motion. A spinning apparatus was set up so that the time for 50 revolutions was recorded for each trial. Each trial was performed 3 times to get 3 time measurements to average for each trial for more accurate results. After getting these results, the centripetal force was calculated using the equation F_{c }= 4 π^{2}mn^{2}r and compared to the force measured in each trial. The percent difference was then derived for each trial.

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