Moment of inertia/distribution of mass [conceptual/experimental]

[anti404]

for our lab last Thursday, we inclined a plane at 5deg, and rolled down two objects, a solid disk and a hollow cylinder, and recorded their position in time with a ultrasonic sensor and the helpful World in Motion software. with this we calculated the object's velocity at certain intervals in Excel, and then calculated these velocity values squared.
we then plotted these v^2 values(y axis) versus the position at which they occurred, or the distance from the sensor at which they occurred(x axis). we then found the slope of this graph.

now, we are somehow supposed to use the following equation to find the so called "Beta value" and compare it to the actual "Beta values", which represents the distribution of masses.
equation: v^2=[2gsin(angle)/(1+Beta)]d+2[Etot/M(1+Beta)].
but honestly, I have no idea how we are supposed to do so --> note, our professor is literally an entire chapter behind the lab. in our actual lecture class, so I've no idea what's going on here.

for the above equation, we have the angle value, know g[ravity], determined the masses, and have the slope of the v^2 vs. position graph.
however, I've got no idea what this slope value represents, as far as our equation is concerned. Likewise, if we are to solve for Beta in the above equation, I wouldn't know how to find Etot, or the total energy of the system. but my biggest question is what that slope value actually represents, because that may make everything far simpler to understand.

phew. that looks as a gibberish mess, but if anyone could possibly provide me with any understanding of these concepts, I would greatly appreciate it - maybe even enough to bake you a(n) (imaginary) cookie.

thanks for any insight, or for just looking,
Justin

 

[ehild]

When a disk, cylinder, sphere of radius R rolls with speed v (this is the speed of the centre of mass) the angular velocity is w=v/R . Both translation and rotation contribute to the kinetic energy.

The rotational KE is RE = 1/2 I w^2. For a body of axial symmetry and homogeneous mass distribution,

$$I=\beta M R^2$$

and w=v/R in case of rolling, so

$$RE=0.5 (\beta M R^2) v^2/R^2 = 0.5 \beta M v^2$$

The total KE is the sum of the translational KE and the rotational energy:

$$KE=0.5 Mv^2+ 0.5 \beta M v^2= 0.5 M(1+\beta ) v^2$$

So the KE of a rolling object is as if its mass was increased because of its moment of inertia. The total energy is the sum of the KE and the potential energy, PE=Mgh. You can express the height with x and the angle of the slope.

ehild