Torque is a measure of how much a force acting on an object causes that object to rotate. The object rotates about an axis, which we will call the pivot point, and will label 'O'. We will call the force 'F'. The distance from the pivot point to the point where the force acts is called the moment arm, and is denoted by 'r'. Note that this distance, 'r', is also a vector, and points from the axis of rotation to the point where the force acts.
Torque is defined as
= r x F = r F sin(
)Angular acceleration can be defined by
or 
For two-dimensional rotational motion, Newton's second law can be adapted to describe the relation between torque and angular acceleration:
is the total torque exerted on the body, and 
is the mass moment of inertia of the body.Part 1
For this experiment the apparatus we set up is comprised of disks that have marks on the edge that is read through the Pasco rotational sensor and we can use logger pro to extract angular velocity data and time. There different sized disk, as well as pulleys and we will be using different amount of hanging masses to collect data. Also for one of the six trials we are suppose to do, a motion detector will be plugged in so that we may compare velocity of the hanging mass versus angular velocity of the disk as well as acceleration of the hanging mass versus the angular acceleration of the disk.
The apparatus will bounce up and down as the hanging mass falls and is retracted by the pulley and disk. Logger pro will then develop graphs of angular velocity over time. We will then take the slope from when the mass falls and the mass rises. Summing both quantities from the slopes will give us an average angular acceleration.
Above image is hanging mass of 25 g.
Small pulley with diameter of 2.602 cm and mass 10 g.
Steel disk with diameter 13.515 cm and mass 1361 g.
Above image is hanging mass of 50 g.
Small pulley with diameter of 2.602 cm and mass 10 g.
Steel disk with diameter 13.515 cm and mass 1361 g.
Above image is hanging mass of 75 g.
Small pulley with diameter of 2.602 cm and mass 10 g.
Steel disk with diameter 13.515 cm and mass 1361 g.
Above image is hanging mass of 25 g.
Large pulley with diameter of 5.00 cm and mass 20 g.
Steel disk with diameter 13.515 cm and mass 1361 g.
Above image is hanging mass of 25 g.
Large pulley with diameter of 5.00 cm and mass 20 g.
Steel aluminum with diameter 13.515 cm and mass 466 g.
Above image is hanging mass of 25 g.
Large pulley with diameter of 5.00 cm and mass 20 g.
Steel disk with diameter 13.515 cm and mass 1361 g.
We engaged the bottom disk to rotate adding a mass of 1348 g with the same diameter of 13.515 cm.
Above image is a table of our data we recorded.
Quick Summary
Logically we can see that angular acceleration is increased when more mass is added to the force of torque. With a pulley doubled in size, trial 1 compared to trial 4, our value in trial 4 shows us that the larger the radius less acceleration there is. When we engaged the bottom steel disk in rotation, trial 6, the acceleration of the system significantly dropped. More weight centered at the disk that is rotating will cause the system to become slower. Versus when we add more weight to the hanging mass will cause the system to increase in acceleration.
Part 1.2
Here we added a motion detector so that we may accomplish comparing angular velocity with velocity and angular acceleration with acceleration.
As before the Pasco rotational sensor will develop a graph so that we may find angular velocity and angular acceleration. Simultaneously, as the Pasco rotational sensor gives us that data we will also have the motion sensor collecting data so that we may find velocity and acceleration of the hanging mass.
Above image is how our apparatus was set up.
Above image is our graphs that logger pro developed.
Above image is the avg of the apparatus's fall and rise of the hanging mass and the rotation of the disk for angular velocity, angular acceleration, velocity, and acceleration.
To compare the two we will use these formulas.
& 
of the disk is 7.04 rad/s. Calculating with the hanging mass over radius is 6.85 rad/s.There is a 2.8 percent error between the measured and the calculated
.α of the disk is 3.47 rad/s/s. Calculating α with the hanging mass over radius is 3.36 rad/s/s.
There is a 3.2 percent error between the measured and calculated α.
Part 2
With the data collected we will be able to figure out the moment of inertia of each of the trials.
First we come up with a model to find the moments of inertia in which we will compare with the perpendicular inertia of the system.
Below is our measured inertia versus the perpendicular inertia from the disks.
Final Summary
Part one it seemed our data made sense. As angular acceleration was increasing when we were adding weight to the hanging mass. When we changed the mass of the disks from lightest to heaviest angular acceleration went slower the heavier the disks were.
Errors for experiment 1.2 with the motion detector yielded promising results. As the comparison from angular velocity versus velocity and angular acceleration versus acceleration gave of low error values between the two. Its hard to say why an error would exist as the experiment ran both the rotational sensor and the motion sensor were collecting at the same time. It is possible that the disk sensor was slightly collecting data faster as the motion detector.
Part two, data for the moment of inertia are not even close to the projected 2 percent or 47 percent error. I retyped in the calculator our values at least ten times. Which is not saying much for the error is our own some how. I know the amount of air plays a role in giving us a frictionless disk during the experiments. During each trial i know we were turning off and on the air. That could be a factor. Trial 1 and 4 we would have hoped that the average angular acceleration in both the value would double. We verified the setting on logger pro and ran trials over again. It is a mystery to why the numbers we calculated from the measured values do not match.
Assuming all went well the measured values we plug into the expression that was derived would have came close to the perpendicular moment of inertia. Telling us that the amount of force required to rotate that disk is equivalent to the speed and acceleration of the falling hanging mass.
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