Physics of Theatre ProjectPhysics of Theatre Project

Center of Center of MassMass

or or Why Personnel Lifts Stand UpWhy Personnel Lifts Stand Up

and Why They Fall Downand Why They Fall Down

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Who We AreWho We AreEric C. Martell, PhD

Associate Professor and Chairof Physics and Astronomy

Millikin University, Decatur IL

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Verda Beth Martell, MFA

Opera Technical DirectorKrannert Center for the Performing

Arts

Assistant Professor of TheatreUniversity of Illinois at Urbana-Champaign

Technical Director Physicist

What We’ll Talk AboutWhat We’ll Talk About• What makes something stable.• Many techniques to find the center of

mass/gravity for an object.• Lots of ways to fall off of ladders.• Why you should use your outriggers.• How dynamic movement figures into stability.• Why the footer should not be the kid who is easily

distracted.

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How?How?• Math

o A little more intensive than past sessions. We will post this PowerPoint on our website (Google “Physics of Theatre”) and on the USITT app.

• Demoso Meet Ernesto – He has balance issues.

• Graphicso We’ve generated a few AutoCAD drawings to illustrate our

models.

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It’s about StabilityIt’s about Stability• Stability is a simple thing.

o If the center of mass is over the base, it is stable. o If the center of mass is not over the base, it is unstable.

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What is the Center of What is the Center of MassMass

• The point where half the mass is in front, half behind, half above, half below, half to the left, and half to the right.

• “Average” position of all the mass.• Does not need to be a point that’s part of

the object – consider a donut.• Center of Mass vs. Center of Gravity

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Finding the Center of Finding the Center of MassMass

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Example – Finding CMExample – Finding CM• Center of Mass of a flat

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Example – Finding CMExample – Finding CM• Break the flat up into rectangular sections, each

with a readily identifiable CM:

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Example – Finding CMExample – Finding CM• Make a table of the x and y coordinates and

weights/masses of each piece (using an average weight density of 1.1 lb/ft2 for ¼” lauan on a 1x3 pine frame).

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Example - CalculationsExample - Calculations

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Example – Checking Example – Checking ResultsResults

• We found xCM=7.4 ft and yCM=4.1 ft.

• Actual center of flat

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Using ExcelUsing Excel

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No Party in the GenieNo Party in the Genie

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Hanging MethodHanging Method• Only works for Homogenous materials.• Cut out the profile.• Hang from a point and draw a line straight down.• Hang from a different point. Draw a line straight

down.• Where the lines cross is the center of mass.

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Dynamic LoadsDynamic Loads• As performers, stagehands, etc, move around on

scenery, Newton’s 3rd Law tells us that whatever forces it applies to them (support, helping them walk/run, helping them stop), they apply back to it.

• Those forces cause torques, which can cause objects to tilt, and if strong enough, tip over.

• When we’re concerned: when the torques caused by the dynamic loads are larger than the “stabilizing” torques holding object in place (gravity, screws/bolts…).

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Dynamic LoadsDynamic Loads• What kind of forces are we talking about?• If a person is moving at initial speed v, and they

stop in a time interval t, they will have an acceleration of a=v/t. The force needed to stop them will have magnitude F=ma, or F=mv/t.

• These forces can be as large or larger than the weight of the person.

• What effect can these forces have?• Spreadsheet

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Force Generated by One 200 lb Person Stopping Abruptly

v (ft/s) t (s) a (m/s2) m (slug) F (lb)Gentle 1 0.1 10 6.21 62.1Moderate 2 0.1 20 6.21 124.2Walking 4 0.1 40 6.21 248.8

What can you do to increase What can you do to increase

stability?stability?• Widen the base.

o Add outriggerso Make the whole object larger

• Effectively widen the base or resist the toppling forceo Guy wireso Stairs

• Make the base heavier to lower the combined center of gravityo Person on ladder baseo Hang sandbagso Add stageweights

• Restrict the movement of the object or of people climbing on the object.o Railingso Harnesseso 3 points of contacto Tie into another objecto Trap your movable object between other objects.

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Dynamic Loads - Dynamic Loads - WagonsWagons

• Let’s say you’ve got something moving on a wagon (great-grandma’s haunted antique armoire) which travels onstage and then comes to a stop. If stopped too suddenly, it can tip (just like you on a train).

• What causes it to tip? Newton’s 1st Law of Motion – An object in motion will remain in motion until acted upon by an outside force. In this case, there is an outside force – the friction between the base of the armoire and the wagon.

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Dynamic Loads - Dynamic Loads - ExampleExample

• Can pivot around front corner.• How big can a be without tipping?

o Left end of base cannot lift off wagon.

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va

fs

Dynamic Loads - Dynamic Loads - ExampleExample

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fs Fg (acts at CM)

FN

• When accelerating, FN no longer acts at center – position depends on acceleration.

• If it doesn’t tip, net torque=0 (around CM).

Dynamic Loads - Dynamic Loads - ExampleExample

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fs Fg (acts at CM)• Torque = Force*Lever Arm (=rFsin)• For weight, lever arm=0, torque=0.

FN

Dynamic Loads - Dynamic Loads - ExampleExample

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rN

FN

rs

• s=fs*rs

• N=FN*rN

• If it’s not tipping, s=N

• FN=mg• fs=ma• rs= height of CM=yCM

• rN=horizontal distance from CM

fs

Dynamic Loads - Dynamic Loads - ExampleExample

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rN

FN

rs

• If it’s not tipping, s=N

• ma(rs)=mg(rN)

• Furthest over FN can shift: the far right edge (rN=xCM).

• a=(xCM/yCM)*g

• If a is bigger than this, it will tip!

fs

xCM

yCM

Walking up a flatWalking up a flat

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