Rolling Motion

This video-based lab involves the study of several uniform objects of circular cross-sectional area. They roll down without slipping from the top of an incline. We wish to find the acceleration of each of them, using the conservation of energy.

1. Pre-lab Activities

• Let’s race a ring, a solid sphere, and a solid cylinder, down a ramp in pure rolling. Which one will win the race? Watch this video.

Link (Links to an external site.)

• What is the physics behind it?
  • First, review the concept of the rotational inertia. Watch this video:Link (Links to an external site.)

• • Now, review the derivation of the rotational inertia for a solid cylinder (from Ch10): Icm=12mR2$I_{cm}=\frac{1}{2}m{R}^2$. For a ring (empty disk), all of its mass is on the outer surface, a distance R form the cm. Therefore Icm=mR2$I_{cm}=m{R}^2$, etc.

• • Now that you know the values of the rotational inertias, watch this video to see the reason why they will roll at different speeds:Link (Links to an external site.)

• • Here is a more detailed analysis that gives the acceleration of the solid cylinder:Link (Links to an external site.)
  • Now repeat the same derivation for the general expression of the linear acceleration of common objects with circular cross-sectional area. The only difference is the X-factor in front of the expression for the moment of inertia for each type of object: Icm=X(mR2)$I_{cm}=X\left(m{R}^2\right)$. You should get

acm=gsinθ1+X,$a_{cm}=\frac{g\sin \theta }{1+X},$

where θ$\theta$ is the angle of inclination of the ramp.

• I$I^{}$In our lab, we analyze the video recordings of several objects with circular cross-sectional area as they roll down the incline without slipping. Logon to Pivot Interactives, and launch the lab, Moment of Inertia of Rolling Cylinders: Water-Filled Disk. Play the video and watch the disk rolling down the incline. If we measure the time t$t$ it takes for the disk to cover a certain distance x$x$ on the ramp from zero initial speed, then x=12at2$x=\frac{1}{2}a{t}^2$, from which we can find the experimental value of a$a$ by fitting the curve of x$x$ vs t$t$.

2. The Lab

• In Pivot Interactives, go to the lab, Moment of Inertia of Rolling Cylinders: Water-Filled Disk, and choose Full Disk, Close-up. Play the first video to see the details of the disk. This will help you determine the mass distribution of the disk, and hence the proper formula to use for its rotational inertia.
• Now switch to Data Shot at the bottom of the video screen. Deploy a scale, and rotate it to align with the orientation of the ramp. Also add a timer, and a protractor to measure the angle of inclination, θ$\theta$. See the screenshot below.

• Play the video, and pause it at various moments to record the distance x$x$ the disk has moved along and ramp, and the time t$t$. Get a minimum of 6 sets of data (x,t)$(x,t)$, before the disk reaches the bottom of the ramp.
• Repeat the same process for three additional configurations: Full disk (frozen), Empty disk, and Half-full disk.

3. The Lab Report

This is a formal, group lab report. It should include the following parts:

• Coverpage
• Equipment (no serial numbers are needed)
• Procedures
• Data: The angle of inclination of the ramp, and a set of (x,t)$(x,t)$ for each of the four configurations.
• Analysis:
  • Determine which of the four configurations is/are example(s) of pure rolling of a rigid object, to which the theories above applies. For each such case, plot x$x$ vs t$t$, and fit it into a quadratic function. The theoretical curve is given by x=12at2$x=\frac{1}{2}a{t}^2$. Use this to determine aexp$a_{exp}$, and compare with the corresponding value of ath$a_{th}$ obtained form the formula ath=gsinθ/(1+X)$a_{th}=g\sin \theta /(1+X)$, with g=9.80m/s2$g=9.80{m/s}^2$. Find the percent error.
  • For the case(s) which does not/do not satisfy the condition of pure rolling of a rigid object, find aexp$a_{exp}$with the same method as above and discuss qualitatively why you think the motion is faster or slower than a corresponding rigid object in pure rolling — for example, compare the cases of a disk full of liquid water vs a frozen disk. Do the experimental results agree with your qualitative prediction?

• Discussion: Possible sources of errors, etc.

Please save the lab report as a PDF or Word document, and upload it.