Physics of Theatre Project
Center of Mass
or
Why Personnel Lifts Stand Up
and Why They Fall Down
4/13/2015
1
Who We Are
Verda Beth Martell, MFA
Technical Director
Opera Technical Director
Krannert Center for the Performing Arts
Eric C. Martell, PhD
Physicist
Associate Professor and Chair
of Physics and Astronomy
Millikin University, Decatur IL
Assistant Professor of Theatre
University of Illinois at Urbana-Champaign
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What 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?
• 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.
• Demos
o Meet Ernesto – He has balance issues.
• Graphics
o We’ve generated a few AutoCAD drawings to illustrate our
models.
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It’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 Mass
• 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 Mass
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Example – Finding CM
• Center of Mass of a flat
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Example – Finding CM
• Break the flat up into rectangular sections, each
with a readily identifiable CM:
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Example – 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 - Calculations
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Example – Checking Results
• We found xCM=7.4 ft and yCM=4.1 ft.
• Actual center of flat
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Using Excel
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VectorWorks
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No Party in the Genie
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Hanging 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 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 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.
Force Generated by One 200 lb Person Stopping Abruptly
v (ft/s)
a (m/s2)
t (s)
m (slug)
F (lb)
Gentle
1
0.1
10
6.21
62.1
Moderate
2
0.1
20
6.21
124.2
Walking
4
0.1
40
6.21
248.8
• What effect can these forces have?
• Spreadsheet
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What can you do to increase stability?
• Widen the base.
o
o
Add outriggers
Make the whole object larger
o
o
Guy wires
Stairs
o
o
o
Person on ladder base
Hang sandbags
Add stageweights
o
o
o
o
o
Railings
Harnesses
3 points of contact
Tie into another object
Trap your movable object between other objects.
• Effectively widen the base or resist the toppling force
• Make the base heavier to lower the combined center of
gravity
• Restrict the movement of the object or of people climbing on
the object.
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Dynamic Loads - Wagons
• 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 - Example
a
v
fs
• 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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Dynamic Loads - Example
FN
fs
Fg (acts at CM)
• When accelerating, FN no longer acts at center –
position depends on acceleration.
• If it doesn’t tip, net torque=0 (around CM).
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Dynamic Loads - Example
FN
fs
Fg (acts at CM)
• Torque = Force*Lever Arm (t=rFsinq)
• For weight, lever arm=0, torque=0.
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Dynamic Loads - Example
FN
rs
rN f s
•
•
•
•
•
•
•
ts=fs*rs
tN=FN*rN
If it’s not tipping, ts=tN
FN=mg
fs=ma
rs= height of CM=yCM
rN=horizontal distance from CM
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Dynamic Loads - Example
FN
xCM
rs
yCM
rN fs
•
•
•
•
•
If it’s not tipping, ts=tN
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!
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Walking up a flat
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