11-May-2015: Moment of Inertia and Frictional Torque

Purpose: Use appropriate measurements to calculate the total moment of inertia and frictional torque in the system,  and use their relationship to calculate the time it takes a cart to travel 1 meter.

Procedure: Given a large metal disk on a central shaft, we were to measure the mass, length, and radius of the disk and shaft in order to find the total moment of inertia. The mass of the entire disk and shaft was labeled in grams on the side but we needed the mass of each individual piece. Given that both parts were of the same density, we solved for the volumes of each part to figure out the percentage of the total mass each part was. Once we found the mass of each piece, we were able to solve for the total moment of inertia by adding up each individual moment of inertia. The inertia of each cylinder is given as  I = 1/2*M*R^2

Calculations for total moment of inertia

Calculating for total moment of inertia, we got I = 0.021 Nm.

Then, we were to solve for the frictional torque acting the apparatus. The trick to this part was finding an easy way to measure the angular acceleration of the disk needed for the calculation. By using video capture to record a specific amount of rotations the disk made over a period of time, we were able to plot a graph of theta over time. 

graph of theta vs time from video capture.

Solving for the 2nd derivative of the function of theta vs. time gives us the angular acceleration of the apparatus.

Calculations for angular acceleration of disk

Our calculations gave us Alpha = -1.1158 rad/sec^2.

Next, we connected the apparatus to a 500 gram dynamic cart by winding it with a string. The cart is placed on an inclined track so that it can roll down a distance of 1 meter. We recorded the angle of the incline of the track to be 58.1 degrees from the horizontal. Our objective from here was to use our measurements of the total moments of inertia and angular acceleration to calculate how long it should take for the cart to travel 1 meter from rest, and compare our result to the actual experimental trails.

Cart on inclined track attached to disk/shaft apparatus

Below is our calculations for finding the amount of time it should take, theoretically, for the cart to travel 1 meter. In order to find out how long it takes, we first had to find out the cart's acceleration, and before finding that, we had to create a set of equations using Newton's second law. We have our calculation for frictional torque using T(frictional) = I*Alpha. We drew our rigid body diagram of the system in order to write an equation for our forces in the x and y direction. Having 2 unknowns, T(radial) and the acceleration of the cart, we had to solve for T(radial) using the equation for net torque, T(radial) - T(frictional) = I*Alpha .  Using the relationship between angular acceleration and linear acceleration Alpha = a/r , we can substitute to get a singular equation solving for the acceleration of the cart. Finally, using kinematics we can use the equation distance = 1/2*a*t^2 to solve for time.

Calculations for time (t) it should take for cart to travel 1 meter down incline.

Results: The calculated result for the amount of time it should take the cart to travel 1 meter down the track came out to 8.58 seconds. We then used a stop watch to measure how long it actually took for the cart to travel 1 meter in three trials. The average of the three trials came out to 8.04 seconds.  The average of our measured results was 6.3% off of the calculated result.

Conclusion/Uncertainty: By measuring specific characteristics of the disk/shaft apparatus, we were able to solve for its total moment of inertia. Then, by using a video capture method to solve for angular acceleration, we were able to solve for the frictional torque of the apparatus. Finally, by using those values and understanding of Newton's second law, we were able to solve for the acceleration of the cart and eventually the time it took the cart to travel 1 meter. Although we were not within the 4% mark, it was still reasonably close. There were a few aspects that could have skewed our final results including the human error on the stopwatch, friction between the cart and the track, and the propagated uncertainty compounded from our measuring utensils.