The Physics of the Adhesive Duck Deficiency

By Beth Shore

I love the show the Big Bang Theory, and in it there is a lot of physics both
shown and talked about. In one episode, Penny falls in her shower and dislocates
her shoulder. This, according to Sheldon, was due to the lack of adhesive ducks on
the tub.

http://www.youtube.com/watch?v=gtUVmx5VXR4

So I wondered, just how important are adhesive ducks (or any other stickers) in preventing shower injuries? By how much do they increase the friction coefficient, providing a safer shower experience? The ASTM F-462 (American Society of Testing and Materials) requires that tubs have a static coefficient of friction of at least 0.04 1. Assuming Penny weighs 55.0 kg, we can calculate the angle at which her force due to gravity.

The vertical components, since she is not accelerating in this direction, sum to 0. 

Fn - Fg cosθ = 0
Fn = Fg cosθ

And, since we don’t want her to slip in the x direction either, we set this
acceleration equal to 0 as well. This is a situation known as static equilibrium.
Fgsinθ - Ffriction = 0
Fgsinθ = Ffriction

Ffriction = μ Fn
Fgsinθ = μ Fg cosθ
sinθ = μ cosθ

http://www.permasafetytub.com/faq.htm

sinθ / cosθ = μ
μ = tan θ
tan θ = 0.04
θ = 2.29o

This isn’t a very large angle, thereby making the shower a very dangerous place without anything to increase the coefficient of friction. There are no specifications for the amount that adhesive stickers, such as the whimsical ducks Sheldon talks about, increase the static coefficient of friction. So, what would the coefficient need to be so that Penny could take a large step in the shower, say 30o, without falling? Using what we found before, tan (30) = 0.58. This is a significant increase from the static coefficient in a tub without such protective measures.

Next, I decided to look at the force that was exerted on Penny’s shoulder to
dislocate it. We know that the moment of intertia of a long uniform rod rotating about the end is 1/3 ML2. Assuming Penny is 5’6’’ and weighs 120 pounds, this comes out to be 1/3 (55 kg) (1.7m)2 = I
I = 53 kg m2
T= I α sin θ

Because her initial angular velocity was 0, and we can assume the fall took approximately 1 second, and she rotated 90 degrees, we can calculate the angular acceleration using the equation Δθ=ωot + (1/2) α t2
Π - π/2 = (0 rad/s)(1s) + (1/2) α (1 s)2
π/2 * 2 = α
α = π rad/s2

Using our torque equation, T= I α sin θ
T= (53 kg m2) (π rad/s2) sin (π/2)
T = 167 Nm

To figure out force, we have to refer to the equation
T = r F sin θ
167 Nm = (.85 m) F
F = 196 N

Which is a very large force. So, had Penny added adhesive ducks to her hower floor; she could have saved her shoulder a lot of force from the fall.