LAB 6: Rotational Dynamics

LAB 6: Rotational Dynamics

OBJECTIVES

To investigate and understand moment of inertia as it relates to rotational motion.

To relate angular and linear position, velocity, acceleration, and kinetic energy.

INVESTIGATION 1: MOMENT OF INERTIA

Bodies resist all attempts to change their motion, whether that be to accelerate them in a straight line, as we have been experimenting with so far, or to set them spinning. The opposition to linear acceleration is simply the object’s mass; however, in its reluctance to be accelerated into circular motion, a more complex variable, Moment of Inertia comes into play. Depending on an object’s shape its moment of inertia can vary widely.

The general form of the equation for moment of inertia is

For this equation to be usable, you must find the equation to transform dm into dr which is called for by the particular body you are working with. Often, this takes the form of a density equation or a relation of how much mass sits at any radius from the center of mass. For example, for an object of uniform area density,

where m is the mass and A is the area of the disk. Let’s take the case of a circular disk. The dark circle in the middle, with radius r and breadth dr, will have area dA, where

Substituting in:

We simplify to get dm in terms of dr:

which allows us to calculate the integral.
Question 1-1: Finish the derivation above for the moment of inertia of a disk. What units doeshave?

The above equation holds when the object is spinning about its center of mass. When the axis of rotation is offset by a distance d from the center of mass, moment of inertia is calculated by

There is a table of the moments of inertia of various common shapes in your textbook (or see internet).

Moment of inertia functions asthe “mass” variable in the equations which govern rotation. In this investigation, we will be using a version of Newton’s Second Law which describes force and acceleration in a rotational setup:

This will allow us to measure moment of inertia directly from a known torque and angular acceleration.

You will need the following materials for this investigation:

PASCO rotation setup

rectangular mass, metal hoop

Hanging mass

Force probe

Masking tape

Activity 1-1: Measuring Moment of Inertia

The rotation setup has two platters. The main platter has a step pulley in the center of it, which you will wrap a string around to exert force on the setup. The mass of the whole platter is

Mass(whole platter) = 0.991 kg

Find the radius of the platter.

The step pulley in the center of the platter will add to its moment of inertia. Because it is a complex structure, you will not have to predict or directly measure this; use the following values instead:

Mass of step pulley: 97 g

Moment of Inertia of step pulley: 2.2 x 10-4 kg m2

Prediction 1-2: Subtract the mass of the step pulley from the mass of the whole main platter and predict the moment of inertia of the remaining disk.

Add the step pulley’s moment of inertia to this value to find the total moment inertia of the main platter.

Tie the string to one of the small holes on the step pulley, and wrap the string around the largest step of the step pulley, which has a radius of 2.50 cm. Tie the other end to the force probe.
Open the experiment file Moment of Inertia 1 from the Physics 131 folder. The setup should include one force probe set to “pull positive” and one photogate set to “photogate and picket fence”.

The platter has been set up to work with a photogate to measure its rotation. There are two (or four)tabs evenly spaced around the edge of the platter to break the photogate beam.
Zero the force probe. Press Start, then begin pulling the force probe so that the platter accelerates as the string unrolls. IT IS IMPERATIVE THAT THE FORCE YOU USE TO PULL THE PLATTER BE AS CONSTANT AS POSSIBLE. This may take several tries.
When you have succeeded in attaining a nearly-constant force over at least 10photogate measurements, look at the acceleration-time graph. If your force is constant, your acceleration should be nearly so also. (There will always be a slight variance because of the imprecision of the measuring setup.) The more measurements you can get a constant force for, the better your data will be. If the acceleration graph disturbs you, you can always find the slope of the velocity vs. time graph.
Use Fit or Statistics to find the average acceleration during these measurements.

Print out a copy of your graph to turn in at the end of lab.

Question 1-2: Find your torque by multiplying your force (in N) by the radius of the step pulley (in m). Why do you use the radius of the pulley and not the radius of the disk?

Now findusing

Does this agree with your prediction? Explain any discrepancy.

Activity 1-2: Measuring Moment of Inertia with different torques

To get a constant force for this experiment you will be using a hanging mass bob (wooden block). Find the mass of the bob:

Calculate the weight of the bob.

Attach one end of the string to the mass bob and wrap the other end around the largest of the steps of the step pulley. Then hang the string over the pulley so that the bob is free to accelerate toward the floor.
Use both the main platter and the auxiliary platter for this experiment.

(Mass(aux platter)=0.894 kg)

Question 1-3: Calculate the torque which the mass bob will exert on the platter:

Use the same experiment file. You only need to graph acceleration.
Start the experiment and allow the bob to free-fall toward the floor.
Use Fit or Statistics to find the average acceleration while the mass bob was falling:

Now use your calculated values forandto find the moment of inertia of the whole setup:

Repeat this experiment using the remaining two steps of the step pulley (radii 2.00 cm and 1.50 cm). Calculate the torque exerted on each radius. Then use your experimental data to calculate I for each radius. Comment on the I’s you find: should they be the same? Why?

Print out a copy of your graphs to turn in at the end of lab. Each radius’ experiment should have its own sheet.

Question 1-4: How well do your three measurements of I agree? Explain any discrepancies.

Activity 1-3: Offset Axes and Complex Objects

In this activity you will test the rule that says that if you have a complex object and know the moment of inertia of each part, you can combine them with simple addition to find the total moment of inertia of the object.

You will also test the equation given in the overview of this investigation for moment of inertia when the axis of rotation is offset from the center of mass.

You have already measured the moment of inertia of the rotation setup’s main disk. Now you will predict and measure the moment of inertia which results from adding masses to the main disk in various shapes and positions.

The first additional mass you will use is a metal hoop. Use a ruler to measure the interior and exterior diameters and multiply by ½ to find the internal and external radii:

The hoop’s mass is:

0.701 kg

Now use the following formula to calculate the hoop’s moment of inertia:

Prediction 1-2: What should the total moment of inertia be if the hoop is put on the disk so that its center of mass coincides with the axis of spin of the disk setup?

Test this prediction. Using the same setup that you used in Activity 1-2, find the angular acceleration which results from your known torque. You need only use one of the steps of the step pulley.

Question 1-5: Was your prediction correct? If not, can you explain any discrepancies?

Print out a copy of your graph to turn in at the end of lab.

Next you will offset the hoop so that one of its edges is at the spin axis. Calculate the new moment of inertia for the offset hoop:

Run the experiment again with the hoop offset and determine the moment of inertia.

Question 1-6: Was your prediction correct? If not, explain any discrepancies.

Print out a copy of your graph to turn in at the end of lab.
Next you will use the supplementary rectangular mass with a hole in the center. Its mass is

0.690 kg

Prediction 1-3: Calculate the moment of inertia if the square mass is spun about the hole in the center. Remember to add the moment of inertia of the main disk to that of the square mass. You can find an appropriate formula in your text for the rectangular mass.

Run the experiment with the mass on top of the disk. Record the angular acceleration you measure below and use it to calculate the moment of inertia:

Print out a copy of your graph to turn in at the end of lab.

Question 1-7: Was your prediction of the moment of inertia correct? Explain any discrepancies.

INVESTIGATION 2: CONSERVATION OF ENERGY IN ROTATIONAL SYSTEMS

As you have learned in lecture, the analogies between the equations of linear and angular dynamics are very strong. Virtually anywhere you see “v” in linear dynamics,can be inserted in an angular calculation. Likewise, “m” becomes. “a” becomes.

It is not surprising, then, when the kinetic energy of a rotating system is found to be

In this investigation we will verify that this formula provides correct conservation of mechanical energy.

In addition to the materials that you used in Investigation 1, you will need:

A motion sensor to watch the mass bob as it falls.

Activity 2-1: Conversion of Gravitational Potential to Rotational Kinetic Energy

You have already been using theof a falling mass bob to accelerate your rotation setup. Now we will go back and look at this experiment more precisely.

Place the motion sensor on the floor and rotate the sensor head so it is looking up at the mass bob. Plug it into the PASport interface and select position and velocity to be read out.
Use the single platter.

Determine where you are going to start your mass bob. Allow the mass bob to fall attached to the string just like you did in the previous activities, but this time recording its position with the motion sensor. Notice that when the bob gets within about 20cm of the motion sensor, it stops reading properly. Use the span of valid distance measurements to set the scale over which you will measure the gravitational potential energy. Enter the total span you can measure as the total change of height of the mass bob:

Question 2-1: Find the gravitational potential energy of the mass bob at the top of its trajectory relative to the end of the motion sensor measurement, which will function as the bottom of its fall.

Open the experiment file Rotational Kinetic Energy from the desktop. You should have a photogate set to “Picket Fence” as before, as well as a motion sensor. Add the motion sensor if necessary.
You should use the top step of the step pulley.

Prediction 2-1: Using, predict the angular acceleration of the disk. Use the values ofand which you found at the beginning of the lab in the investigation with the single platter.

Calculate through what angle the disk will rotate when the mass bob has traveled a vertical distance h (hint, how many revolutions will the disk make when the string spools out h?):

Calculate the time it will take the accelerating disk to cover this angle:

Calculate the angular velocity of the disk at time t:

Calculate the downward velocity of the mass bob at time t:

Calculate the kinetic energies of the disk and mass bob at time t:

Is this kinetic energy equal to the potential energy you calculated earlier?

Run the experiment. You may wish to do a practice run first to make sure that the mass bob can be seen by the motion sensor throughout its entire descent.
Analyze your data. Use the angle, angular velocity, and angular acceleration data from the photogate to determine experimental values to compare with your predictions:

Print out a copy of your graphs to turn in at the end of lab.

Question 2-3: How well did your data agree with your predictions? Make sure you explain any discrepancies.

Question 2-4: Was energy conserved in this experiment? Is this what you expected?

End-of-lab Checklist:

Make sure you turn in:

your lab manual sheets with all questions answered and all relevant data filled in.

Graphs from Activities 1-1, 1-2 (3 graphs), 1-3 (3 graphs), and 2-1