blog 13

Moment of inetia 

In this lab, we had a large disk shaped object with two small cylinders attached on either side of its center of mass, pivoted on a (presumably) frictionless axis so that it spins vertically. Our goal was to figure our the acceleration of a cart attached to a string wrapped around this object which is released to run down a fictionless ramp.

Pictured, our apparatus without cart attached

Since we were going to be using torque=m(alpha) forms of newtons second law, it becomes immediately necessary to calculate the moment of inertia of our spinning object. We are given the mass of the object, but we cannot use a simple 1/2mr^2 equation to determine I, because of the small side cylinders.  The calculation is shown below

M=4.615

dimentions of large disk

R=0.101m

H=0.015m

V1=(pi)R^2(H)= 4.809*10^-4

And of each small cylinder

R=0.0157

H=0.0516

V2=(pi)R^2(H)= 3.647*10^-5

Total volume of the system=Vt=V1+2(V2)

Therefore, mass of each component (where M1 is the large cylinder and M2/M3 are the small side cylinders):

M1=M*(V1/Vt)=3.915

M2 and M3=M*(V2/Vt)=0.35

From these we can determine the moment of inertia for the system

moment of inertia system = moment of inertia large cylinder-2(moment of inertia small cylinder)

I=1/2(M1)R1^2+2(1/2M2R2^2)

I = 2.008*10^-2

Using this value we can now approximate alpha by plugging into an appropriate equation

ma=sin(x)mg-t   t=m(g-a)   t=m(sin(x)g-alpha(r))

t(r)=I(alpha) 

a=r(alpha)

mr(sin(x)g-alpha(r))=I(alpha)

mgrsin(x)=alpha(I+mr^2)

alpha=mgrsin(x)/(I+mr^2)

plugging our values for m, g, r and I, we get the value

alpha=-1.253 rad/s^2

Now that we have a theoretical baseline, we must run the experiment to see how close we came in our prediction. We run the experiment and timed the result. Giving us a value for alpha of

alpha=-1.266

This value is only 1.04 percent off of our predicted value, Which means that our model is fairly accurate. The small amount of error is probably due to small errors in measuring the dimensions of our spinning apparatus, and the fact that we treated the ramp as frictionless.