blog 12

Rotation 

In this lab, we have a system which pumps air between two metal disks, allowing the one on top to rotate with nearly 0 friction. We set up a system with a string wrapped around this top disk, which then goes over a pulley to a hanging mass, as pictured below Our goal was to determine how certain variables within this system affect the angular acceleration of the spinning disk.

We run 6 trails during this experiment. The first three use a top disk (the one that will spin) made of steel, with a pulley of equal diameter, but increasing weight of the hanging mass (0.0245, 0.0495, 0.0995 respectively). For the next two experiments, we use the same hanging mass and a larger diameter pulley, but change the material of spinning disk from steel in trail 4 to aluminum in trial 5. Finally, in trial 6, we use 2 steel disks stuck together as our rotating object. In all cases we measured the acceleration of the system by way of a motion detector mounted below the hanging mass. When the mass reached the bottom of its string, the inertia in the disk cause it to start to rise, so in our graphs for acceleration we ended up with positive and negative values for acceleration. We chose to call these a up and a down, and averaged them in the following equation.

a_avg=((a_up^2)^1/2+(a_down^2)^1/2)/2

We then equated a_avg to average angular acceleration by the equation

alpha = a_avg/r

where alpha is angular acceleration and r is the radius.

In order to figure our how angular acceleration was theoretically going to be affected in each situation, we had to derive an equation for angular acceleration. To do this, we used T=I(alpha)

and F=Ma equations as shown below.

T=I(alpha)

T=t(r)

I=1/2mR^2

Ma= Mg-t

a=alpha(r)

from these we get

Mr(alpha)= Mg-t

t=M(g-alpha(r))

Mr(g-alpha(r))=I(alpha)

Mrg=alpha(I+Mr^2)

alpha=Mrg/(I+Mr^2)

From this, we can see that in theory, the larger the hanging mass the greater the alpha, and the higher the inertia of the spinning mass the lower the alpha. Finally, the larger the r the greater the alpha. 

Below we have the data from our 6 trials

As we can see, our predictions hold out fairly well. Trail 1, 2 and 3 show increasing alphas and increasing mass. Trials 4 and 5 show that the lower moment of inertia spinning disk of the 2 (aluminium) has a larger alpha than the steel disk trial. In addition, when comparing trails 1 and 4, the only difference is the size of the radius of the pulley, and the trail with the larger pulley (trial 4) has a higher alpha. Finally, in trail 6, with the highest moment of inertia for any trial and the lowest hanging weight, we have our lowest value for alpha.