- Find the relationship between the angular speed and the angle formed by the apparatus
- Using equation found in part 1 to calculate several theoretical values of angular speed responding to different angles.
- Measure the period and use it with other known condition to find the experimental value of angular speed.
- Compare experimental values and theoretical values of angular speed.
Set-up:
How to set up the apparatus of the experiment:
1. Mounting an electric motor on a surveying tripod.
2. A long shaft going vertically up from the shaft.
3. Mounting a horizontal rod on the vertical rod
4. A long string is tied at the end of the horizontal rod.
5. A rubber stopper at the end of the string.
6. A ring stand with a horizontal piece of paper sticking out.
Figure 1: The apparatus of the experiment.
How the experiment is performed:
1. Turn on the electric motor to let the string spin.
2. As the motor spins at a certain angular speed w, the mass revolves around the central shaft at a certain radius and the angle.
3. Performing 6 trials with an increasing amount of voltage each time and collecting the data.
4. The data needs collected each trial is the time that the motor need to travel in a certain number of revolutions and the height of the horizontal paper attached to the ring stand as the motor grazes it when the motor spins.
The diagram is shown below help you imagine easier what we are going to measure and collect.
Figure 2: The diagram shows how equipments are set up in this experiment.
Data Collection:- Professor performs the first trial to show us how the system works, he then lets us measure necessary quantities such as length of the string, height of tripod, etc...to calculate the angular speed and check whether it makes sense.
Figure 3: Professor is performing the first "checking" trail. - If the result we get make sense, we will start doing real trials with increasing speed of motor, results in a wider angle, and higher height which is measured by the ring stand.
Figure 4: Performing real trials with different speed of motor, different angle and height. - We will do 6 trials and record appropriate data.
Data Analysis:
Since our objective is to find out the relationship between the angle and omega (⦶ and w), we need to come up with an equation to solve for angle and omega.
As we look at the diagram, we notice that the string make with the height of the tripod an angle that we can find thanks to the right triangle. The height is equal the difference of the height of the surveying tripod and the height of a horizontal piece of paper which the rubber stopper hits.
After finding an equation to solve for the angle, we come up with another equation to solve for omega as a function of the angle and measured data (height and length of string).
Below is the derivation of an equation for the angle and omega.
Figure 3: Deriving an equation to solve for the angle and omega.
Since we have both equation, what we need is data to plug in. We performed six trials in this experiment with different angle and period at each time. For each time, we increase the power of the electric motor, the angular speed (w) increase, the mass revolves around the central shaft at a larger radius and the angle also increase. Figure 4 is our data.
Figure 4: Height and time are recorded for each trial in order to find out the angle and angular speed.
We then plug our data into our equation to calculate the angle and the angular speed. Below is an example how we calculated them. (Figure 5)
Figure 5: Calculating the angle and the theoretical value of angular speed.
We keep doing the same process for the next 5 trials, and here are our results.
Figure 6: Results for the angle and angular speed after calculating all six trials.
One of the data we also recorded in this experiment is the time that is required to finish one revolution for each trial. With that data, we are able to find the experimental value of angular speed through an equation:
w = 2∏ / T
Example of calculating the experimental value of angular speed.
Figure 7: Calculating the experimental value of angular speed based on the time we recorded.
We perform the same process for the next five trials. After we're done, we put them into the same table with the angle and theoretical value of angular speed. The table is shown in figure 8.
Figure 8: The results of experimental value of angular speed along with these previous results.
Since we have the experimental and theoretical value of angular speed, we enter them into the Logger Pro and graph them to check how well we did with this experiment. If we obtain the slope of the graph equal 1, this means that our experimental values match with our theoretical values. In case the slope is not exactly equal 1, this refers that we did something wrong in the experiment or made wrong assumption.
So we go and enter the data into the Logger Pro.
Figure 9: Entering experimental and theoretical value of angular speed into Logger Pro.
Then we linearize the data and look for the value of the slope of the graph.
Figure 10: Graph showing the relationship between experimental and theoretical value of angular speed.
m = 1.005
As we get the value of the slope m = 1.005, we can conclude that our experimental values almost perfectly match with the theoretical value of angular speed. This also means that we follow the procedure exactly and perform a great experiment.
Conclusion:
In this experiment, our objective is to find the relationship between the angle and the angular speed. We found that angular speed is related to the angle through an equation in which the angle is calculated thanks to the right triangle with the hypotenuse L and height H-h.
The slope of the graph between experimental and theoretical value of angular speed is:
m = 1.005
Although the slope is not exactly equal 1 like our expectation, the result is not far off from 1, which indicates that our results are pretty good.
The slope is not exactly equal 1 because of some of the following reasons.
1. We couldn't record exactly the time for the the string finished one revolution, thus the error may interfere with our experimental value of angular speed.
2. We may make unnecessary rounding number when calculating the theoretical value of angular speed.
3. The uncertainty in measuring the height of the surveying tripod, the ring stand with the horizontal paper, and the length of the string also affect our results.
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