Lab 18: Moment of Inertia and Frictional Torque
Joel Cook
Nina Song
Eric Chong
Lab finished May 22, 2017
In this lab, we are attempting to calculate the moment of inertia and angular acceleration for a rotational apparatus. With the moment of inertia and angular acceleration, we can predict the time it will take a cart, tied to the apparatus, to descend down a ramp.
Theory: By measuring the mass and dimensions of an object, we can use this data to calculate the moment of inertia of a disk with the following equation:
I (moment of inertia) = 1/2 * m (mass) * r^2 (radius)
Our apparatus is a series of disks (cylinders) and the total moment of inertia of an object is the sum of the moments of inertia of each part of the object.
To predict the time the cart will take to descend a ramp of known angle and length, we measured the angular deceleration of apparatus and used kinematics to calculate the time it should take:
Angular acceleration = tangential acceleration
radius
Tangential acceleration = acceleration of the string = acceleration of the cart
Displacement = initial velocity * time + 1/2 * tangential acceleration * time^2
Procedure:
The rotational apparatus for the lab is pictured below:
We used calipers to measure the dimensions of the rotating portion of the apparatus. The dimensions are shown below:
The mass of the rotating portion is stamped on the disc. The dimensions of the object were used to calculate the volume of each portion of the apparatus. Assuming a uniform density, the volume of each portion can be used as a percentage of the total volume to calculate the mass of each part. The mass of each part, and dimensions, were used to calculate the moment of inertia of each component.
For the next portion, we used slow motion video capture and logger pro on the computer to record video of the disc as it rotated and slowed. The video was then analyzed to determine the tangential speed of the disc as it slowed. The edge of the tape on the disc was used as a reference point to click in each frame of the video, as shown below.
The video analysis yielded the graph below for X and Y axis position vs time for the rotation.
To calculate angular acceleration (α), the tangential velocity was calculated and graphed by knowing that tangential velocity is:
Vtan = (Vx^2 + Vy^2)^(1/2)
The tangential velocity was used to calculate ω through the following relationship:
Vtan = ω* r (radius)
By graphing angular speed vs time, we can calculate the angular acceleration of the disc by finding the slope of the line through the data points graphed, as shown below:
Once angular acceleration was determined, and the relationship between angular acceleration and tangential acceleration identified (α = a/r), the acceleration of the cart was known and the time for the cart to travel one meter down the track shown below could be calculated.
The string was wrapped around the smaller axis of the apparatus and timed as it descended down the ramp. This value was compared to the value obtained through calculation.
Data/Analysis:
As described previously, the volume of the rotating portion of the apparatus was calculated from the dimensions of the apparatus.
The volume for each small cylinder was calculated:
V (cylinder) = pi * radius^2 * height
V = pi *(30.8/2)^2 * 52.3 = 38966.64 mm^3
The volume of the disc was calculated:
V = pi*r^2*h = pi * (200.46/2)^2 * 14.6 = 460784.8 mm^3
Adding together the disk and the two cylinders gives a total volume of 538718.13 mm^3. The percentage of volume that each part contributes was calculated:
Volume of part/total volume * 100 = % Volume of apparatus
Each cylinder was approximately 7.23% and the disk was approximately 85.45% of the total volume. Therefore, multiplying the total mass of the object by the percentage each part contributes, the mass of each part can be determined:
Total mass * % volume = mass of each part
The mass of each cylinder was approximately 333.665 g and the mass of the disk was 3947.67 g.
With the dimensions of each part and the mass of each part, we can calculate the moment of inertia:
I (moment of inertia = 1/2*Mass*radius^2
I (total) = I (disk)+I (cylinder 1) + I (cylinder 2)
I (total) = 1/2*(0.333665 kg)*(0.0154 m)^2 + 1/2*(0.333665 kg)*(0.0154 m)^2 + 1/2*(3.94767 kg)(0.100023 m)^2
I (apparatus)= 0.0199 kgm^2
The video analysis described in the procedure yielded an angular acceleration of -0.1903 rad/s^2. This is the force of friction on the disc as it rotates.
Scene 2, The Cart:
As shown in the following calculations, the moment of inertia was used to calculate the acceleration of the cart.
The first photo shows the free body diagram for the system:
By using Newton's second law, force equals the product of mass and acceleration and therefore torque equals the product of moment of inertia and angular acceleration, we can calculate the acceleration of the cart. The angular acceleration from friction was accounted for to more accurately predict the acceleration.
This acceleration for the cart was then used to predict the time it should take for the cart to descend the track. The prediction used a 40 degree angle for the track so the actual trials were setup with a 40 degree angle as well.
Δ x = Vo*t + 1/2*a*t^2 (Vo = 0)
t = (2*Δx/a)^(1/2)
t = (2*1/0.0402)^(1/2) = 7.05 seconds
Three trials were timed for the cart descending the ramp. The data was as follows:
Trial 1: 7.36 s
Trial 2: 7.65 s
Trial 3: 7.63 s
Average time: 7.55 s
Difference between predicted and actual: (1- 7.05/7.55) * 100 = 6.62%
Conclusion:
Even though the actual and predicted time for the cart descending the track was more than 6% different, I would classify the lab as a success. We were able to take basic measurements and use the data to make a fairly accurate prediction. Errors and uncertainties existed at multiple points throughout the lab. The measurements for dimensions and mass, as well ass the angle of the track, have a small uncertainty that was mitigated as much as possible by using more accurate implements to measure. There is also an uncertainty in the timing of the descent of the cart that are inevitable due to the unreliable nature of human reflexes.
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