Physics of Theatre Project Center of Mass or Why Personnel Lifts Stand Up and Why They Fall Down 5/4/2015 1

Who We Are Eric C. Martell, PhD Associate Professor and Chair of Physics and Astronomy Millikin University, Decatur IL 5/4/ Verda Beth Martell, MFA Opera Technical Director Krannert Center for the Performing Arts Assistant Professor of Theatre University of Illinois at Urbana-Champaign Technical Director Physicist

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. 5/4/2015 3

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. 5/4/2015 4

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. 5/4/2015 5

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 5/4/2015 6

Finding the Center of Mass 5/4/2015 7

Example – Finding CM Center of Mass of a flat 5/4/2015 8

Example – Finding CM Break the flat up into rectangular sections, each with a readily identifiable CM: 5/4/2015 9

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/ft 2 for ¼” lauan on a 1x3 pine frame). 5/4/

Example - Calculations 5/4/

Example – Checking Results We found x CM =7.4 ft and y CM =4.1 ft. Actual center of flat 5/4/

Using Excel 5/4/

VectorWorks 5/4/

No Party in the Genie 5/4/

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. 5/4/

Dynamic Loads As performers, stagehands, etc, move around on scenery, Newton’s 3 rd 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…). 5/4/

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. What effect can these forces have? Spreadsheet 5/4/ Force Generated by One 200 lb Person Stopping Abruptly v (ft/s)t (s) a (m/s 2 )m (slug)F (lb) Gentle Moderate Walking

What can you do to increase stability? Widen the base. o Add outriggers o Make the whole object larger Effectively widen the base or resist the toppling force o Guy wires o Stairs Make the base heavier to lower the combined center of gravity o Person on ladder base o Hang sandbags o Add stageweights Restrict the movement of the object or of people climbing on the object. o Railings o Harnesses o 3 points of contact o Tie into another object o Trap your movable object between other objects. 5/4/

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 1 st 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. 5/4/

Dynamic Loads - Example Can pivot around front corner. How big can a be without tipping? o Left end of base cannot lift off wagon. 5/4/ v a fsfs

Dynamic Loads - Example 5/4/ fsfs F g (acts at CM) FNFN When accelerating, F N no longer acts at center – position depends on acceleration. If it doesn’t tip, net torque=0 (around CM).

Dynamic Loads - Example 5/4/ fsfs F g (acts at CM) Torque = Force*Lever Arm (  =rFsin  ) For weight, lever arm=0, torque=0. FNFN

Dynamic Loads - Example 5/4/ rNrN FNFN rsrs  s =f s *r s  N =F N *r N If it’s not tipping,  s =  N F N =mg f s =ma r s = height of CM=y CM r N =horizontal distance from CM fsfs

Dynamic Loads - Example 5/4/ rNrN FNFN rsrs If it’s not tipping,  s =  N ma(r s )=mg(r N ) Furthest over F N can shift: the far right edge (r N =x CM ). a=(x CM /y CM )*g If a is bigger than this, it will tip! fsfs x CM y CM

Walking up a flat 5/4/