Exploration

Use an on-line simulation from the University of Colorado PhET group to investigate the relationships between angular acceleration, moment of inertia, angular momentum and torque.
Link to the simulation: http://phet.colorado.edu/en/simulations/torque

This is a simulation written in Java.  Here are instructions on running Java PhET simulation on a Windows or macOS computer.
If you cannot rum Java on your computer, run the simulation via Cheerpj (the browser compatible version).  It takes a long time to load, and it runs slower, but it works.

Click the Intro tab explore the interface.

• Use your mouse to apply a torque to the wheel for a short time interval Δt to give the wheel an angular impulse and set it spinning.
• Slow the rotation rate with the brake.
• Watch the velocity and acceleration arrows for the bugs.
Describe the direction of those arrows, while the angular speed of the wheel is increasing, constant, or decreasing.

Click the Torque tab.

• Set the applied force equal to 1.5 N.
• Click Go let the simulation run for approximately 10 seconds.
• What is the magnitude and direction of the torque on the wheel?
• What happens to the lady bug?
• What provides the centripetal force to keep the bug moving in a circle?
• Why does this eventually fail?

Click the Moment of Inertia tab.

• Choose radians for the angular units.
• Set the scale of the torque graph to show a range of -20 to 20 Nm.
• Set the scale of the Moment of Inertia graph to show a range of -2 kg m² to 2 kg m².
• Set the angular acceleration graph to show -20 rad/s² to 20 rad/s².
• Put the mouse finger anywhere on the boundary between the green and pink circles.
• Hold down the left mouse button and move mouse.  Look at the graph and try to apply a force that creates a torque of between 5 and 15 Nm.
• Use the ruler to measure the radius r of the boundary between the green and pink circles.  Record r in your log.
• Calculate what the tangential component of the applied force must have been.  Record F_tang in your log.
• Compare the torque τ and the angular acceleration α and calculate the moment of inertia I of the disk from τ = Iα.  Record I in your log.
• Compare with the moment of inertia displayed in the graph.  Record the comparison in your log.

Click the Angular Momentum tab.

• Choose radians for the angular units.
• Set the scale of the Moment of Inertia graph to show a range of -2 kg m² to 2 kg m².
• Set the angular speed to be 1 rad/s.  This fixes the angular momentum unless you apply a torque.
• While the disk is moving, change the inner radius to 2 m.  What happens to the moment of inertia and the angular velocity?
• Make some more changes to the inner radius, outer radius and mass of the disk.  Describe what happens.

Exploration Deliverables: (to be included in the your journal)

• Visuals:  Screenshot of the Intro tab showing the wheel, the bugs, the velocity and acceleration arrows, while the wheel is spinning, braking, and coasting.
  Screenshot from the torque tab showing the applied force, the direction of torque, the motion of the ladybug, and the acceleration vector.
  Screenshot from the moment of inertia tab showing the torque graph, the angular acceleration graph, the moment of inertia graph, and the disk with inner/outer radii visible.
  Screenshot from the angular momentum tab showing the angular speed, the moment of inertia, and changes after adjusting radii or mass.

• Analysis: 
  Intro tab analysis:  Describe the direction of velocity and acceleration arrows when the angular speed is increasing, constant, or decreasing.  Explain why the acceleration arrow behaves differently from the velocity arrow.
  
  Torque tab analysis:  State the magnitude and direction of the torque for a 1.5 N applied force.  Explain what happens to the ladybug and why.  Identify what provides the centripetal force keeping the bug in circular motion.  Explain why the centripetal force eventually fails.
  For the 0.5 N force describe how the acceleration vector changes over time.  Explain whether the acceleration vector can ever point directly toward the center.  When braking is applied describe how the acceleration vector responds and why.
  
  Moment of Inertia tab analysis:  Record the measured radius r.  Compute the tangential force component from the measured torque.  Use this to compute the experimental moment of inertia.  Compare the computed value with the graph’s displayed moment of inertia.  Explain why the moment of inertia graph stays constant even when the force changes.
  
  Angular Momentum tab analysis:  Describe what happens to the moment of inertia and the angular velocity when the inner radius is changed while the disk is spinning.  Explain how this demonstrates conservation of angular momentum.  Describe what happens when mass, inner radius, or outer radius are changed.

• Results:  Based on your exploration, summarize how velocity and acceleration arrows behave in different rotational regimes, how torque affects rotational motion, and how mass distribution affects moment of inertia.

Experiment 2

Take a meter stick with a clip to which you can attach a weight.  Grab the stick on one end, and hold it horizontal, with the weight close to your hand.  Then have someone slide the weight out to other end.  Does it become harder to hold the stick horizontal?  Why?  You are holding up the same weight.

In this experiment, we will use the meter stick as an artificial forearm.

The arm (excluding the shoulder and wrist) is composed to two major segments.  The upper arm is attached to the shoulder.  The forearm is attached to the upper arm at the elbow.  Let us only concentrate on situations where the upper arm is in the vertical position.  Then the forearm can move in two directions, upwards or downwards.

The upper-arm and the forearm can be thought of as two rigid levers, joined at the elbow, which acts as a pivot point.  Muscles attached to these levers provide the force required to articulate their motion.  Since muscles can only provide force by contraction, they must always work in pairs.  As the contracting muscle, (red), tightens, it applies a force to the arm.  At the same time, the opposing muscle, (black), relaxes, thus allowing the arm to move.  We have a pair of simple levers.

When we lift a load with our biceps muscle, this muscle does positive work.

The triceps muscle does positive work when we push down on something.

There is also a third force on the forearm, the force the upper arm's bone exerts on the forearm at the elbow joint.  This force does no work, because the elbow joint is not moving.

AI Study Tip:

Example prompt for lab preparation:  'In my physics lab, we are modeling the human forearm as a lever.  The elbow is the pivot, the biceps provides an upward force, and a weight is held in the hand.  If the arm is in static equilibrium, help me set up the sum of torques equation.  How do I account for the weight of the forearm itself acting at its center of mass?'

Procedure:

Attach 3 clips to the meter stick, one at each end, and one in the middle as shown in the figure below.  Hang the stick from the force sensor.  Adjust the position of the center clip so the stick is horizontal.  Use the force sensor and Capstone to measure the weight of the stick and the clips.  (Always press the tare button on the force sensor before you attach something to the hook to make a measurement.)

Move the center clip to a position 20 - 25 cm away from the left edge of the stick.  Now the stick is no longer horizontal.  You have to push down on the left end of the stick to bring the stick to a horizontal position.  When you push down on the stick, what happens to the reading of the force sensor?

To find out how hard you have to push down to bring the stick to a horizontal position.  Attach a mass hanger to the clip on the left end.  Load masses onto the mass hanger until the stick is horizontal.  The combined weight of the masses and hanger equals the force with which you have to push down.

Fill in the table below.  Record only magnitudes.  (Before taking a force reading, remove the stick, press the tare button, and reattach the stick.)

weight of meter stick and clips
weight suspended from left clip
(optional) weight suspended from right clip
force sensor reading when the stick is again horizontal
distance d from left clip to force sensor hook (pivot point)
distance from force sensor hook to CM of meter stick
distance from force sensor hook to right clip

Experiment 2 Deliverables: (to be included in the your journal)

• Table:  Table of experimental results.

• Analysis:  Modeling the meter stick as a forearm and the force sensor as the biceps, compare the force that the biceps has to exert to keep the forearm horizontal to the force it has to exert to just support the weight of the forearm.  Comment on the relative magnitude of these forces.
  
  List all the forces (magnitude and direction) acting on the forearm (meter stick) and calculate all the torques (magnitude and direction) exerted by those forces about the pivot point, when the stick is horizontal and in equilibrium.  Do all the  torques cancel out?

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Convert your log into a lab report.

Name:
E-mail address:

Laboratory 6 Report

• Make sure you completed the entire lab and answered all parts.  Make sure you show your work and inserted and properly labeled relevant tables and plots in your journal.
• Add a summary reflection at the end of your report in a short essay format.

Save your Word document (your name_lab6.docx), go to Canvas, Assignments, Lab 6, and submit your document.