11-May-2015: Moment of inertia and frictional torque

     The purpose of this experiment was to predict the time it would take for a cart to travel some distance down a ramp by applying Newton's 2nd Law and torque/inertia relationships.
     The apparatus used for this experiment is shown below. It was a symmetrical mass with a large center disk and smaller cylinders on each side of it. The entire thing was able to spin about its center.

One end of a string was tied to one of the smaller cylinders and wrapped around it. Connected to the other end of the string was a cart. The cart was placed on a ramp, which rested on a counter making some angle with the horizontal. Below is a picture of the whole setup. The cart was released and the time it took to travel 1 meter was recorded. This was repeated a few times to get a rough number for the time.

The actual experiment was short and sweet but the theoretical calculations took a little more time. So, lets take a look at those because that is what we need to compare the experimental value with.

     First, the moment of inertia of the whole apparatus was needed. Since it was made up of cylinders, the equation (1/2)MR^2 was used. The inertia of the individual parts were added to find the total inertia. In order to find the mass and radius, though, measurements were made with vernier calipers and the mass was derived using a volume relationship. Below were the steps taken. 

     

     The final step was to derive an equation for the time it would take the cart to travel down the ramp. Newton's laws were applied to the cart and torque relationships were applied to the apparatus. Manipulating the equations gave an expression for linear acceleration. Once linear acceleration was found, playing with kinematic equations allowed time to be solved for in terms of distance traveled and acceleration. Below is the derivation.

     The theoretical calculations produced a time of 9.21 seconds for the cart to travel 1 meter down the ramp. The experiment yielded a consistent time of about 9.70 seconds. A percent error of 5.3%. This error could have been caused friction from the cart and ramp or the string not being parallel to the ramp.