The apparatus is comprised of a large center disk in the middle and the sides are cylinders serving as an axis point of rotation or otherwise known as a central shaft.
We need to know the frictional torque of this apparatus. To do so we need to find inertia and angular velocity.
Image above- our net external is going to be frictional
To find inertia we will take measurements of each of the side cylinders and middle disk. Data that needs to be recorded is the diameter and the lengths/thickness.
With these values we are going to determine the volume of each piece. Each volume piece divided by the total volume will give us a percentage of mass for each piece of the apparatus.
Multiplying each percentage to the total mass will yield the appropriate individual mass of each piece.
Using the formula above with the mass we have and the measured radius we can find each of the pieces inertia and add them for the total inertia of the system.
Now to find angular acceleration was tricky. There are a few methods in which at the time is hard to understand and somewhat still is. One you can take the x and y components and use that to find angular acceleration. Second take a point and mark every rotation and time at which it returns to the same spot.
We took the video capture route and are going to attempt to find angular acceleration that way.
We made sure the camera lens was close to center with the apparatus rotational axis.
On the disk we taped a point in which we plot each time it made one rotation in logger pro.
We took the data and plotted a graph on excel to gain a graph that will give us angular acceleration.
The equation on the excel graph is mimicking the above equation 3.
-0.2168 is equivalent to 1/2 α
Put everything together and our frictional torque comes out to be -0.0092634 N m
Noted in the calculation image we tried to go the route of x and y components but found the method confusing and difficult for us. So we crossed out the attempted calculations. What you see for inertia and angular acceleration was the method described above in which we used to obtain our frictional torque.
Now we can move on the the experiment and predict the time a cart rolls down an incline with this apparatus.
Above image is how the set up of the experiment looks like.
Drawing a free body diagram will give us force equations combined with torque equations we can predict the time it will take for the cart to roll down 1 meter. The circled equation will find acceleration in which we will use in a kinematics equation to obtain the predicted time.
Then we ran the experiment and timed with a stop watch exactly how fast it takes the cart to go down the incline. We did this a few times to gain an average we can use to compare with the predict time.
The times from the stop watch were 8.34 s, 8.22 s, and 8.41 s. Giving us an average time of 8.32 s.
Error between the predicted time and the measured is 0.4 percent.
Conclusion- Combing the methods of Newton's second law, torque, moments of inertia, rotational, and kinematics assisted in predicting the time the carts goes down 1 meter. Problem solving this lab was a bit tricky for finding the angular acceleration because there was a few approaches that could have been taken. We originally attempted to find angular acceleration with the x and y components but was lost in the calculations. We had to switch the method to a way in which we could better understand the process of obtaining angular acceleration. Using the video capture and plotting a dot every time the disk made one rotation until it stopped was better for us. Understanding that the negative sign meant deceleration for angular acceleration. But when we plugged into the equation solving for acceleration the friction is being subtracted from tension times radius of the apparatus.
Errors that played a role could have been a bit from the carts wheels and the track, that friction. Not having the string completely parallel with the inclined track might have a minor impact that was not account for in the expression. The measured uncertainty from the equipment is another reason from the error.
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