Measure the system's angular acceleration, moment of inertia, and derive an equation to find frictional torque in the metal disk system using these measurements.
Experiment:
- The apparatus is composed of 3 cylinders. On it, there's a stapled mark that indicates the total of the 3 cylinders combined (in grams). Record this number.
-Use these measurements to calculate moment of inertia of each cylinder, then the total inertia.
- To determine the angular acceleration, set up video capture to record to the disk after it's given a spin and slows down on its own due to friction.
- Place a tape on tip of disk so we could add new point series later during video analysis. In Logger pro open video capture and set it to record to 15-20 seconds. Go to camera settings, "Adjustments", "Image", and adjust contrast and lighting accordingly. Adjust the camera visibility to have the radius of the large disk in the middle. Test out by start recording video. Make sure that the tape is seen all the time.
- For the experiment, give the disk a spin, then record video. The video should still record until the disk stop on its own.
Data and Analysis:
Calculating Moment of Inertia
- Since only the total mass is given and we can assume that the cylinders have uniform volume density, we can find each of their individual mass.
- We use volume of cylinder = πR2H, and the relationship of ρ=m/v . Note that we need m1+m3 together and m2 alone.
Calculating frictional torque (τf )
- After given a spin, the only torque acting on the system is frictional torque. So for the net torque equation, we can write: Στ: τf = (I-total) α
- For video analysis, set origin for x and y-axis (ideally at the center of the disk), set scale of the radius, add new point series and start adding points taking the tape as the point of interest. Continue adding points until the disk stops.
- The point series will give x and y- displacement. From this we could take the derivatives to get x and y-velocity, by going to new calculated column.
- Since we want to relate our data to rotational motion, we could create another new calculated column for angle. From our x and y-position, use arctan("y" / "x") to get the angle.
- Up to this point, we could check to see if our data is reasonable by graphing angle vs. time and take linear fit,
- Although there are plenty of discontinuity in our graphs, we see that angular velocity decreases with time, which is explained by the fact that the disk slows down.
- Next up, create another new calculated column for tangential velocity, that is the linear velocity at the edge of disk. Using Pythagorean Theorem gets us V= sqrt[(V-x)^2 + (V-y)^2]; and we just found V-x and V-y.
- Up to this point, what we're trying to do is to get Logger pro to give us angular acceleration (alpha). And we know that ω vs. t gives us the slope= alpha. To find ω, use the relationship between rotational and translational motion, one of which is ω= V/R. where R= radius of the disk and V is the one obtained from Pythagorean Theorem.
- Our final data table should look like,
- We could also take derivatives 2 times to get alpha from our angle. But seeing that angular acceleration varies so much although it would've stayed constant, we stay with the idea of using slope of ω vs. t.
- Graphing ω vs. t and taking linear fit gives, α = -0. 4246 rad/sec^2,
- So Στ: τf = (I-total) α = (0.02146)(-0.4246) = -0.009118 N.m; where negative sign indicates opposite direction of motion. We take I-total because the whole system is bound to frictional torque.
Disk-Cart system
- After finding the frictional torque, use this to derive an equation for linear acceleration for metal disks-cart system, where cart is attached by a string and wrapped around one of the smaller radius of the 3 cylinders.
- During this set up, make sure the string is parallel to the track and does not slip as it unwinds from the disk. Keep the angle(track with horizontal) unchanged throughout the experiment.
- As usual, we can obtain two equations one from rotational dynamics and another from Newton's Second Law, sum of forces. Since α= (a-center of mass )/radius, replace this and solve for "a-center of mass".
- The calculation below show how we get the equation of linear acceleration for the cart. Following the lab handout, we set our initial conditions to be when angle (of the track with the floor)= 40°, m-cart = 0.500kg, and r = 0.0158m and find that it takes 8.20 seconds for the cart to travel 1 meter down the track.
- τ-string = TR; where R= radius of the disk the string is wrapped around. Since the whole rotating apparatus is acceleration, we take I-total to be on the other side of our net torque equation.
- Using our real data, m-cart = 0.504kg; angle = 47.8°, same radius, Δx = 0.98m, we get t =0.72s
- Experimental data, where Δx = 0.98m (the cart doesn't quite begin at the top nor get to the very bottom of the track)
- trial 1: t = 7.42 s
- trial 2: t = 7.44 s
- trial 3: t = 7.47s
- If we wrap the string around the larger radius and let go of the cart, it will take longer for the cart to get to the bottom of the incline. This observation is made clear in the derived equation. When you increase the radius, the acceleration gets smaller.
Conclusion
In this uniform rotating apparatus, we could find frictional torque using angular acceleration and calculated moment of inertia. This result could be further used to determine the linear acceleration. Assuming constant linear acceleration, we could use kinematics to solve other possible problems given. For part 1, where frictional torque has to be determined, errors include the fact there's drop frames and the metal supporter gets in the way when points series are taken. Besides the poor video quality, there's also usual measurement uncertainty in the diameters. For part 2, we also have measurement uncertainty in angle, cart's mass, and Δx. Within our range of error and uncertainty, we can conclude practicality in our data analysis.
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