Finding the moment of inertia of a uniform triangle

Objective:

• Finding the moment of inertia of a uniform right triangle thin plate around its center of mass, for two perpendicular orientation of the triangle.

Physical thinking:

• Approach:
• We need to find the moment of inertia of a right triangle rotates around the center of mass, but we don't approach to solve the problem from the center of mass. Here is why:
• If we rotate around the center of mass, we will miss some piece of the whole right triangle while integrating.
• The limit for integration is not simple. So, we need another approach to solve this problem
• Now, we can approach the problem from the theorem we learn in class which is called parallel axis theorem. Basically, the theorem states that

• Since finding the moment of inertia when rotating around center of mass takes more work and may not work out, we can shift the pivot point to the edge of the triangle. Based on the parallel axis theorem, if we can find the moment of inertia when the pivot point is at the edge and know the distance from the center of mass to the edge, we can figure out the moment of inertia when the pivot point at center of mass.

First, finding the moment of inertia when the triangle rotates around its height (longer leg).

• Choose a small representative disk dm inside the triangle. 
• Write an expression for dm
• Sum all of small disks of the triangle by integrating from 0 to H.
• When we're all done finding the moment of inertia when the pivot point is at the edge, we can apply the parallel axis theorem above to find the moment of inertia when the pivot point is at the center of mass.
• Here is the work.

Figure 1: Deriving an equation for the moment of inertia of the triangle rotating around its height.

Second, finding the moment of inertia when the triangle rotates around its base (shorter leg).

• Doing the same procedure like above, and below is the work.

Figure 2: Deriving an equation for moment of inertia of the triangle rotating around its base.

Now we start an actual experiment to find out the experimental value of moment of inertia. The goal is to compare the value we got from our calculation and from the experiment to see how well we run the experiment.

Set-up:

• The apparatus of the experiment includes the rotation stand, steel and aluminum disk, hanging mass, torque pulley, triangle holder and triangle, and lastly Logger Pro.
• Putting the aluminum disk on top of the steel disk into the rotation stand, make sure that only the top disk rotate this time by letting the air run through the end. 
• The large torque volley is put on top of aluminum disk and connects to the hanging mass. 
• The triangle holder is used instead of a drop pin to hold the triangle.
• Connect the apparatus to the Logger Pro and set up the rotation sensor to read 200 counts per revolution.

Below is the apparatus

Figure 3: Apparatus of the experiment. 

The left one is when the triangle rotates around its long leg (or height)

The right one is when the triangle rotates around its short leg (or base)

• Also using caliper and balance to measure the mass and dimension of triangle, hanging mass, torque pulley and steel and aluminum disk.

Figure 4: Measuring dimensions of components of apparatus.

Procedure:

• First of all, the mass and dimension of components of apparatus are measured. The mass of the triangle is 456g, its height(longer leg) and base(shorter leg) are in turn 14.936cm and 9.844cm. The hanging mass is 25.0g. The diameter of the large pulley is 5.02cm. The mass of aluminum disk is 466g and its diameter is 12.630cm. The mass of the steel disk is 1348g and its diameter is 12.628cm. Lastly, the mass of the triangle holder is 26g.
• We actually do three trials: the first one is without the triangle, the second one is when the triangle rotates around its shorter legs, and the last one is when the triangle rotates around its longer legs. 
• Turn on the air supplier to let the disk rotate, make sure the hose clamp opens so that the bottom and top disk rotate independently. As the disk rotates, the hanging mass starts going down and goes back up again. Logger Pro starts collecting the data since the hanging mass is released. The graph obtained in Logger Pro will be the graph of angular velocity vs. time. By taking the value of the slope, we can find the angular acceleration. However, to minimize the effect of friction torque on the system, we take the average angular acceleration as the mass goes down and back up. Likewise, we in turn find the the average angular acceleration when the triangle is attached. Below is the graph of angular velocity vs. time of three trials.

Figure 5: Graph of angular velocity vs. time when rotating without triangle. 

Figure 6: Graph of angular velocity vs. time when rotating around the shorter leg (base).

Figure 7: Graph of angular velocity vs. time when rotating around the longer leg (height).

Discussion:

• The equation to calculate the experimental moment of inertia is similar to the one we used in angular acceleration lab, so we don't have to derive one more time. Equation (1) is shown below
  

• The average angular acceleration is calculated by summing the angular acceleration when the hanging mass goes down and goes back up, then divide by two. The values are recorded in the table below

Figure 8: Result table of average angular acceleration.

• Using equation 1, we can calculate the experimental moment of inertia of each trial. Below is an example how to calculate it.

• Result table for experimental moment of inertia.

• Then, by subtracting the moment of inertia of the system with the triangle by the system without the triangle, we can find moment of inertia of the triangle for each orientation. Below is an example how to calculate and result table after we're done.

Figure 9: Result table of theoretical moment of inertia of the triangle.

• The experimental value of moment of inertia was found, so we need to calculate the theoretical moment of inertia to compare the result and check how accurate the experiment went. 
• Using the dimensions of the triangle and two equations derived when applying parallel axis theorem above, plugging in the number and find the experimental moment of inertia. Here is the calculation.

Figure 10: Theoretical moment of inertia when triangle rotating around its height.

Figure 11: Theoretical moment of inertia when triangle rotating around its base.

• Last but not least, we calculate the percent error to check whether the result comes out like what we expected (within 10%). If so, we performed our experiment well and two equation we derived are correct. Here is the formula we will use to calculate the percent error (formula (*))

• Apply the formula (*) to calculate the percent error when triangle rotating around its long leg.

• Apply the formula (*) to calculate the percent error when triangle rotating around its short leg.

• As you can see, our percent errors for both orientations are fairly small and within 10%. This means that we run the experimental very well and two equations we derived are correct.

Some source of errors may happen:

1. Uncertainty in measurement of dimensions of components of the apparatus.
2. Uncertainty in measurement of angular displacement due to the rotational sensor.
3. Friction may affect the angular acceleration even though it is already minimized by taking the average angular acceleration.
4. We may let too much air run through the system than we actually need.

Conclusion:

• In this lab, we perform an experiment to find out the moment of inertia of the triangle about its center of mass rotating both orientation: one is around its height, and the other one is around its base. 
• We first successful come up with two equations to calculate the theoretical values of moment of inertia. 
• When the triangle rotates around its base, we find: 

• When the triangle rotates around its height, we find:

• The percent error of this experiment comes out very well. The percent error of the triangle when rotating around its base is 2.32% and that of the triangle when rotating around its height is 0.76%. Both are small and within 10% regardless of errors may happen during the experiment.