Lab#17: Finding Moment of Inertia of Triangle

Finding Moment of Inertia of Uniform Triangle

Samuel
Kyle
John
11-16-2016

Purpose:

To determine the moment of inertia of a right triangle thin plate around its center of mass, for two perpendicular orientation of the triangle.

Theory:

The parallel axis theorem states that I_parallel axis = I_around cm + M(d_parallel axis displacement)^2. Because the limits of integration are simpler if we calculate the moment of inertia around a vertical end of the triangle, you can calculate that moment of inertia and then get I_cm from I_around cm = I_around one vertical end of the triangle - M(parallel axis displacement)^2

Procedure:

1. Use same experimental setup as Rotational Acceleration lab 

2. Mount the triangle on a holder and disk

3. Use string wrapped around a pulley on top of and attached to the disk 

4. The tension in the string exerts a torque on pulley-disk combination.

5. Measured angular acceleration of the system to determine moment of inertia of the system.

6. Mount triangle onto the disk-pulley-holder system and measure α 

7. Determine the I of the new system

Data:

m = 453.6 grams = 0.4536 kg

thickness = 7.7 mm = 0.0077 m

Base = 98.0 mm = 0.098 m

Height = 146.3 mm = 0.1463 m

(Above data is on rotating disk WITHOUT triangle. Since rotating with less moment arm the, it has more force acting on the rotating object. And that's how the slope is getting more steep than the next 2 graphs)   

( Above data is disk rotating with short triangle. Short triangle is with longer bass and shorter height. Since the moment arm is longer than the other 2, therefore the rotating speed will be slower than other 2.) 

(Above is the graph of higher triangle. In this graph, the slope is between no triangle and short triangle. Since, it's higher on the axis of rotation, so it won't affect on the speed of rotation. Therefore, the angular velocity will be faster than shorter triangle.)

Calculation:

Theoretical value : 1/18* M*B^2
Experimental Value : I(around cm) = I(around one vertical end of the triangle) - M(d_parallel axis displacement)^2

Short base
Theoretical value = 1/18 * M*B^2 = (0.4536*0.098^2)/18 = 2.4*10^-4 = 0.000242
Experimental value = m*g*r*(1/alpha(short triangle) - 1/alpha(no triangle)) = 0.025*0.027*9.8*(0.489-0.453) = 0.000238

In this case
m(mass of pulley) = 0.025 kg
r(radius of pulley) = 0.027 m 
alpha = (1.925+1.772)/2 = 1.8485 rad/s^2
B(base of triangle) = 0.1265 m
M(mass of triangle) = 0.4536 kg

The difference between theoretical value (0.000238) and experimental value(0.000242) is 0.000004. Which is about 1% error. The experimental and theoretical value we had calculated is almost identical.

Long Base

Theoretical value = 1/18 * M*B^2 = (0.4536*0.1463^2)/18 = 0.000539 = 5.39*10^-4
Experimental value = m*g*r*(1/alpha(long triangle) - 1/alpha(no triangle)) = 0.025*9.8*0.027*(0.544-0.453) = 0.000601 = 6.01*10^-4 

In this case
m(mass of pulley) = 0.025 kg
r(radius of pulley) = 0.027 m 
alpha = (2.16+1.92)/2 = 2.04 rad/s^2
B(base of triangle) = 0.1463 m
M(mass of triangle) = 0.4536 kg

The value difference in longer base triangle is 5.39*10^-4 and 6.01*10^-4. The error percentage is about 10% difference. We do not know what cause this big different in calculation. The value should have come out quite similar just like short triangle. 

Conclusion: 

In this experiment, we compare the inertia of no triangle, short triangle and long triangle to see the different it make when rotating on the same disk. However, to perform this experiment, we have made a quite few assumption. First, we assume there's no outside environment force acting in the system(no air resistance, constant gravity...). Secondly, there no mechanical error or resisting force(string is rotating not slipping, table is perfectly balance, there's no frictional force on disk or pulley). We then compare our theoretical value with experimental value, the resultant value are quite similar for shorter triangle, but bigger different for longer triangle. I assume some of our measurement are a bit off, and when all the calculation added together, the round off error accumulate into big different. The difference for longer triangle is about 10% error. But at the end, our prediction on rotating inertia of a triangle is close to what it actually turn out. We have used parallel axis theorem to find how much center of mass has shifted and add it with inertia of triangle. Furthermore, the resultant upward alpha(angular acceleration) is similar downward alpha. Which is a good sign that our angular acceleration are constant.