Lab#18: Moment of Inertia and Frictional Torque

Moment of Inertia and Frictional Torque

Samuel
Kyle
John
11-16-2016

Purpose:

To determine moment of inertia of combined mass and calculate frictional force acting against the rotating mass. 

Theory:

With given mass and the shape of an object, we can find the moment of inertia when object is rotating. Since, object is decreasing its angular velocity consistently. Therefore, we can calculate the fractional force acting on it, even more on coefficient of friction. In addition, with known constant force apply to the disk tangentially, we can calculate/time the velocity of the object with already known distance.

Procedure:

1. Make appropriate measurement of the rotating part of the apparatus and determine its moment of inertia.
2. Spin the apparatus. Use video capture to determine its angular deceleration as it slow down. Calculate the frictional torque acting on the apparatus.
3. You are going to be connecting this apparatus to a 500-gram dynamic cart. The cart will roll down an inclined track for a distance of 1 meter. Calculate how long it should take for the cart to travel 1 meter from rest. Assume for your preliminary calculations that the track is angled at 40 degree.
4. Set up the apparatus. Determine your actual angel measurement. Calculate what the time for the cart to travel 1 meter should be with this actual angle.
5. Run three trial where you measure the time for the cart to travel 1 meter. Be sure that your instructor witnesses at least one of the trial. If your average time is more than 4% off from what you calculated, figure out what went wrong and repeat the steps again until your predictions and your calculations match.

Measurement & Equation:

Big_diameter (diameter of big disk)= 20.05 cm  = 0.2005 m
Dept_ Big (thickness of big disk)= 14.6 mm = 0.0146 m
Small_diameter (small rod diameter)= 30.9 mm = 0.0309 m
Dept_small (dept of small rod)= 5.1 cm = 0.05 m
Mass (mass of both big disk and small rod)= 4615 g = 4.615 kg

w(omega) = sqrt(v_x^2 + v_y^2) / (0.2005/2) (For the purple graph below)
I(inertia) * ⍺(alpha) = 𝓣_friction
V(volume) = 𝜋*r^2 * h

Graph:

(Above is the graph we used to find the torque due to friction and angular deceleration. We first used camera to videotape the rotating disk. Then upload it to LoggerPro using air drop. Next, use video capture to apply dot in every 5 frames. At the end, we got this up and down curve signifying its position and where's it actually is. And the bottom graph shows how much does disk decelerate, we used from 1.3 second and beyond because it become more stable after that time)

Calculation:

Part 1. Find angular deceleration and frictional torque

Before finding angular deceleration, we first need to know how big of portion that each shape is and it's inertia.

We first find volume of each shape.

Disk = 𝜋*r^2 * h = 3.14 * 0.1002^2 * 0.0145 = 4.6*110^-4

Cylinder = 𝜋*r^2 * h = 3.14*0.01545^2*0.051 = 7.64*10^-4

and we got disk is 85.7% and cylinder is 7.1%(each cylinder)

We used the percentage multiply with weight of total mass to find it inertia.

I_cylinder = 1/2*2 * m*r^2 = 0.655*0.01593^2

I_disk = 1/2 * 3.955*.10^2

I(inertia) * ⍺(alpha) = 𝓣

𝓣 = 0.0062

and deceleration is -0.094

Part 2. Find time for traveling for 1 meter with 40 degree angle

Time (Stop watch)

1. 7.16 s

2. 7.22 s

3. 7.43 s

Average time: 7.27 s

Calculation

mgsin𝚹-I a/r -𝓣/r = m*a

(m+I/r^2)*a = m*g*sin𝚹 - 𝓣/r

a = (m*g*sin𝚹 - 𝓣/r) / (m+I/r^2)

a = 0.039571

then we used kinematic equation 

Δx = v*t + 1/2* a*t^2

Δx = 1/2 * a* t^2

1 = 1/2 * (0.039571)*t^2

t = 7.1 second

Conclusion:

In this experiment, we first have to find out inertia for each rotating shape and how big of portion does each shape weight. Then we can find inertia for the total shape when rotating. We know that torque = I * alpha, so we can find frictional torque(because only force acting on it is friction beside gravity) and we can find alpha using camera and LoggerPro. Since we know the inertia and frictional torque and we also know the tension of a cart pulling on a string in a inclined plane. We can use that to find how long does it take to travel certain distance. In this experiment, we make it travel for 1 meter. Furthermore, our calculated time is 7.1 seconds; stop watch time is 7.27 second. In this case, the number is almost identical. The difference might due to static and kinetic friction between wheels and track or human error when stopping stopwatch. At the end, our calculation is pretty decent and close to our prediction.