Purpose: We want to apply a known torque to an object that can rotate, and measure the angular acceleration. Eventually with this torque and angular acceleration data. We can find a measured value for the moment of inertia.
Model: τ=Iα, τ=TL, I=1/2MR^2
τ is the torque on the disk, I is the moment of inertia, T is the tension, M is the mass of the rotating disk, R is the radius of the rotation disk.
Process: First, we measure the diameter and mass of the top steel disk, the bottom steel disk, the top aluminum disk, the smaller torque pulley, and the larger torque pulley. We measure the mass of the hanging mass.
Then, we plug the the pasco rotational sensor into computer and set up the computer. We only neet to discern which is measuring the top disk. Because there is no defined sensor for this rotational apparatus, we need to create something that work with this equipment. We choose the Rotary Motion, and we set the equation in the sensor setting to 200 counts per rotation because there are 200 marks on the top disk.
We turn on the compressed air so that the disk can rotate separately. We can use the pin to let the top disk rotate only or the top disk and the bottem disk rotate together.
In the experiment 1, 2, and 3: Effect of changing the hanging mass.
In the experiment 1 and 4: Effect of changing the radius and which the hanging mass exerts a torque.
In the experiment 4, 5, and 6: Effect of changing the rotating mass.
For each experiment, we get the graph velocity vs. time as the photo showing:
By looking for the slope of the line, we can get the angular acceleration.
We collect the data which is the mass of hanging mass, the kind of torque pulley, the number and kind of rotating disk, and the angular acceleration of going up and down for each experiment.
τ=Iα
In the comparison group 1, 2, and 3:
We change the hanging mass and keep other factors constant.
We get: m1/m2≈α1/α2
m1/m3≈α1/α3
m2/m3≈α2/α3
In the comparison group 1 and 4:
We change the kind of torque pulley which have different radius and keep other factors constant.
We get: r1/r4≈α1/α4
In the comparison group 4, 5, and 6:
We change the mass of the rotation disk and keep other factors constant.
We get: I4/I5≈α5/α4
I4/I6≈α6/α4
I5/I6≈α6/α5
In comparison group 2 and 4:
We change the mass of the hanging mass and the radius of the torque pulley in the same time and keep other factors constant.
We get: (m2/m4)*(r2/r4)≈α2/α4
The number in the comparison group are not the same because the tension is not equal to the weigh of the hanging mass and there are some friction existing between the disk.
Part 2:
We use our data to determine the moment of inertia of the disks or disk combinations from previous experiments.
Because there is frictional torque in the system. The rotating disk is not trully frictionless and there is some mass in the frictionless pulley which we can approximate as a frictional torque. The angular acceleration of the system when the mass is desecending is not the same as when it is ascending.
We make some assumption in this experiments. The frictional torque in the system is independent of direction of angular velocity. The disk are solid whick without a hole in the center. The moments of inertia of the pulley is 1000 times smaller than the disk, so we choose to ingore it.
Then, we use the model and do some calculation:
Tr=Iα
mg-T=ma
a=αr
By these model:
(Iα)/r=mg-m(αr)
I=(mgr)/α-mr^2
We use I=(mgr)/α-mr^2 to find out the experiment moments of inertia of the disk.
We use I=(M*R^2)/2 to find the theory moments of inertia of the disk.
We compare the theory moments of inertia and experiment moments of inertia:
I1≈Itop steel
I2≈Itop steel
I3≈Itop steel
I4≈Itop steel
I5≈Itop aluminum
I6≈Itop steel+Idown steel
Bass on the assumption that we make, we can say the theory and the experiment are close enough.
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