Stopping the Moon, Part 2

(My apologies for the lateness of this post: blame it on snow, ice, a preview, and a long drive!)

Last week, we looked at the physics of moving and stopping a large rolling platform. This week, as promised, I thought we’d take a look at keeping that platform in place when actors are doing the sort of unexpected, dynamic motion you might expect of them on top of it.

To recap: We have a wagon that weighs approximately 900# (last week’s post indicated the wagon was about 500#--unfortunately, I did not have my notes with me while writing the post, and I grossly underestimated the weight of the wagon). Potentially, we’ll have as many as ten people on top of the wagon, doing what they do, weighing, on average, 180#. (I generally find it safest to imagine them doing some kind of mosh-pit improv exercise.) We’ll be using pneumatic “feet” to hold the wagon in place. What we want to calculate is whether or not our pneumatic system can effectively hold the unit stationary while those actors are doing the hand jive on top.

Them’s the brakes

First, let’s examine the pneumatic system. We want to use four individual, rubber-soled feet attached to pneumatic cylinders with a 2 ½” bore. Ideally, to keep the unit stable, we’ll want to apply as much pressure with the cylinders as we can without actually lifting the unit off the ground. First, we should determine the air pressure required to apply a total of 900# of lifting force with the feet, using the formula:

F=PA

where F is the force applied by each cylinder (in this case ¼ of 900#, or 225#)

P is the air pressure

A is the area of the cylinder’s cap, πr², or 4.9in²

In this case, then, the air pressure required will be about 45.9 psi. As the parts of our system are rated for 150 psi, we know we can safely pressurize the system to this level.

Knowing that we can apply 900# of force using the brakes, we can determine how much lateral force would be required to overcome the friction between the brakes and the floor. (For now, we’ll consider the wagon with no actors on it.) As we saw last week, the force to overcome friction between two objects is a function of the normal force (in our case, the downward force applied by the pneumatic cylinders) and the coefficient of friction of the two materials. (Last week we used rolling friction, as we were talking about casters. In this case, we’re talking about static friction. For more on rolling, static, and kinetic friction, see the references below.) The coefficient of friction for a rubber object sliding across painted masonite is 0.8; the cylinders apply 900# of force; therefore to overcome the friction of the rubber-soled feet, we would need to apply 0.8x900, or 720, pounds of lateral force to the wagon.

Jump, jive and wail

Now, there’s still the impact of inertia to consider: 900# of material doesn’t necessarily want to accelerate easily. However, before the wagon will move, we’ll have to apply more than the 720 pounds of lateral force required to overcome the friction between the brakes and the floor.

Knowing this, we can begin to examine what kind of movement would be required on the part of the actors into the side of the wagon to deliver this amount of force. (Notice that we’re still not considering any actors on top of the wagon as yet.) Let’s say that the average 180 pound actor can jump straight up 1 ½ feet, and that it takes 1 second to jump up and return to the ground. (Thanks to one of our directing grads, Justine Muzay, for being my guinea pig for this measurement—and please know I’m reasonably certain she’s much slighter than our average 180# estimate!) Using the equations of constant acceleration, we can determine the rate at which she’s able to accelerate her body by jumping:

x=v₁(t)+1/2(a)(t²)

where t is the time over which acceleration happens, or half the total time of the move, 0.5 sec

x is the distance over which acceleration happens, or half the total distance of the move, 0.75 ft

v₁ is the initial velocity of the unit (in this case, standing still), 0 ft/sec

a is the acceleration, in ft/sec²

Substituting our known values reveals an acceleration of 6 ft/sec². Substituting this value into the equation to determine the force to accelerate a mass, we can determine the force our actor applies to the ground to achieve his or her jump:

F_a=ma

Where F_a is the force applied, expressed in pounds

m is the mass of the object, in slugs (180/32.2 or 5.59 slugs)

a is the acceleration of the object, in ft/sec² (6 ft/sec²)

Substitution reveals that our actor is able to apply about 33.5 pounds of force when jumping from a stationary, standing position.

Bodies in motion

For safety’s sake, let’s round up the amount of force an actor can apply to the wagon by jumping to 100#. (This provides a roughly 3:1 safety ratio over our average estimates, which seems wise.) Intuitively, we know that by jumping straight up, an actor will not impart any lateral force; by jumping perfectly horizontally, they’ll impart 100# of lateral force. Although no actor will ever be able to actually jump sidewise, we can use 100# as a very conservative estimate of the force imparted to the wagon by an actor moving about on top of it. In other words, we’ll estimate—conservatively—that one actor will be able to apply as much as 100# of lateral force to the wagon.

Given this conservative estimate, if there are seven actors on top of the wagon, and they all jump in the same direction at the same time, they’ll impart a maximum of 700# of lateral force—less than the 720# of lateral force required to overcome the friction between the feet and the floor. Although there may be as many as ten actors on the wagon at any time, it is unlikely they will all be jumping in exactly the same direction at exact same instant, that they will be jumping horizontally, and that they will be exerting the full amount of force they are capable of when they jump. Assuming that this last statement is a safe assumption to make, the pneumatic brakes we’ve designed should work just fine for our purposes.

References:

Hendrickson, Alan. Mechanical Design for the Stage. Chicago: Focal Press, 2008.

The Physics of Theatre Project at the University of Illinois