Measuring Moments of Inertia And them some

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Measuring Moments of InertiaAnd them some

• MECH 398 Introductory Lecture for Moments of
  Inertia Lab
• Prepared by Paul Ostic, Sept 2003

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Moments of Inertia Introduction

• Analogous to Mass but for Rotations.
• Fma TIa
• Kmv2/2 KI?2/2
• Axis is important!

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Rotating Mass on a String

• Kinetic Energy
• Torque and Angular Acceleration

O
r
m
F
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Integrate to get Rigid Body

• A rigid body is a connected blob of tiny masses
  on strings
• Integrate

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Mass Distribution

• m1
• r2
• Be careful. Light but big may have a higher
  moment of inertia that heavy but compact!

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Bicycle Wheel vs Crankshaft

• Bicycle Wheel
• Hollow Cylinder
• m1kg
• r0.5m

• Crankshaft
• Solid Cylinder
• m10kg
• r0.1m

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Look in the Textbook

• Axis orientation is important
• Axis location is important
• Parallel Axis Theorem
• Formulae for common shapes

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Experimental

• Measure
• Plug and Chug
• Done?

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__it Happens!

• I cant add
• I cant count
• The tape measure is lousy
• One of the assumptions doesnt apply
• I read the formula wrong
• The lab manual is wrong
• I cant tell the difference between cm and
  inches.
• I used g instead of kg
• I measured to the wrong place
• I punched it in wrong on the calculator
• The 7 key is sticky
• We forgot to divide by 10 for 10 periods
• The x10 button on the scope was in
• You mean that is adjustable?

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Validation of Results

• Know the answer is reasonable so that
• You dont look dumb.
• Your boss pays you.
• You know your assistant did it right.
• The bridge doesnt fall down.
• The cappuccino maker doesnt blow up.

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How to Validate?

• Check the answer in the back.
• Ask a friend.
• Is engineering really democratic?
• Do it a second time.
• Good, I can make the same mistake twice!
• Different result? Do it a third time.
• Hurray for democracy!

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Compare with ???

• Literature
• Manufacturers spec
• Engineering estimate
• Simple experiment
• Scale model
• Limiting case

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Engineering Estimate

• Order of Magnitude
• one or two significant figures
• Over and Underestimates
• Keep it Simple
• Am I really out to lunch?

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Estimation Tools

• Use simple models
• Avoid Calculators
• round if necessary consistent direction
• Write down all steps
• you can retrace them later if things dont match
• Do sketches
• you can confirm measurements and dimensions later
  if things dont match

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Time out for worked examples

• Lawn mower rotor
• Solid cylinder
• Hollow cylinder
• Squash Racquet
• Point mass at end
• Point mass at centre
• Long slender rod

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Validity and Accuracy

• Confirm validity by comparing with a good
  answer or engineering estimate.
• Estimate accuracy using uncertainty propagation.

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Measurement Uncertainty

• READING Uncertainty is /- ½ of smallest division
• MEASUREMENT Uncertainty accounts for
• Experimental technique
• Awkward shapes
• Moving targets
• Poorly defined endpoints
• How far is it to Toronto?

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Uncertainty Exercise

• Using stopwatches built into audience
  wristwatches.
• Start and stop on command
• Holler out values
• Make a quick graph on the board
• Estimate central value and uncertainty

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Uncertainty Propagation
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Uncertainty PropagationTrifilar Table

• Prelab
• How big is each coefficient?
• How accurately does each quantity need to be
  measured?
• What is the uncertainty in the final result?

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Sample Uncertainty Propagation

• Estimated period reading for trifilar table
• Equation 3.2

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Considering Assumptions

• Observe applicability
• rolls without slipping
• What if the assumption does not completely apply?
  (e.g. the oscillations are not infinitesimal)
• Magnitude of effect?
• Mathematical correction?
• Increased uncertainty?

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Small Angles?
The PendulumRich physics from a simple
system Robert A. Nelson and M. G. Olsson, Am. J.
Phys., Vol. 54, No. 2, February 1986
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What did that mean?

• For a mass on a string (simple pendulum), larger
  oscillations have longer periods
• Does this equation work for this lab? (NO!)

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Small Angle Solution

• How do we determine the ideal period?
• Do we need to measure the size of oscillation?
• Read the lab procedure and think about it

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It is a long lab.

• Calculations are all to be done during the lab.
• If you are not prepared in advance, you will not
  finish.
• The report should be concise.

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Experimental

• Prepare!
• Uncertainty equations
• Assumptions
• Review Procedures
• Engineering Estimates
• Measure Accurately
• Plug and Chug
• Values
• Uncertainties
• Validate
• Critical Assessment

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Pre Lab Preparation

• Read the manual
• Make procedural notes
• Prepare graph templates
• Prepare data collection tables
• Work through the uncertainty propagations using
  partial derivatives calculate the coefficients.
• Look up formulae from the textbook
• make sketches for quick reference

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There are a lot of questions!

• Lots of questions are to think about, and dont
  require a firm answer.
• Some questions are very similar but with
  different words.
• If something doesnt make sense, continue on
  until the next checkpoint if you can, then ask
  the TA.
• Dont panic. Have fun.

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Scheduling

• Do the roll oscillation segment first.
• 230 to 430 or 430 to 630
• Do the compound pendulum and trifilar segment
  later.
• 230 to 500 or 400 to 630

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Lab Reports

• Hand in before you leave the lab
• Neat handwriting is okay
• or
• Due a week after completion of the second part of
  the lab.
• Include the date and time that each part of the
  experiment was done
• Must be typed

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Tutorial Exercise to Hand In

• Make engineering estimates of the moment of
  inertia of each component you will be using in
  the lab.
• Use the instructions in the lab manual for
  guidance
• Show all calculations. Do not use a calculator.
• Hand in sketches, computations, and a conclusion
  statement for each component.