A cylindrical soda can has a radius of 4 cm and a height of 12 cm. When the can

Question

A cylindrical soda can has a radius of 4 cm and a height of 12 cm. When the can is full the center of mass is 6 cm above the base of the can. When soda is removed from the can the center of mass drops. However, when the can is empty the mass is also 6 cm above the bottom of the can so at some point, the center of mass would have to rise. Find the depth of the soda in the can for which the center of mass is the lowest. Neglect the mass of the can, and assume the density of the soda is 1 g /cm3 and the density of air is 0.001 g/cm3.

Explanation / Answer

cute problem... since we are neglecting the mass of the can (but including the mass of the air? that seems weird, but you have to go with it...) the idea here is that the center of mass is determined by the amount of air and soda in the can. Obviously if the can has all soda, the cm is at the center of the can (6 cm) and if the can is all air (i.e. empty) then the cm is at the center of the can.

So to find the lowest point of the center of mass, we simply write the position of the center of mass as a function of the height of the soda, then take the derivative of the function and set it equal to zero. Then solve for height of the soda and youre done! You have the height (or depth) of the soda for which the cm is minimum.

Like this:

mass of soda in can = volume of soda * density = Ah*1

mass of air in can = voume of air * density = A(12-h)*0.001

position of the center of the soda = h/2

position of the center of the air = h + (12-h)/2 = 6 + h/2

Now...

x = (1/total mass) * (mass of soda * position of soda +

mass of air * position of air)

or

x = (1/ Ah + A*.012 - A*0.001h) * (Ah * h/2 + 0.001A *(12-h)(6+h/2) )

simplify

x = (1/0.999h + 0.012) * (0.25 h^2 + 0.072 - 0.001*0.25h^2 ) =

= [ 0.072 + 0.999*0.25*h^2 ] / (0.999h + 0.012)

take derivative, dx/dh

dx/dh = ( 2*0.999*.25 h ) (0.999 h + 0.012) - 0.999 * (0.072 + 0.999*0.25*h^2)

simplify, set equal to zero

0 = 0.499 h^2 + 0.005994 h - 0.2495 h^2 - 0.071928

This is a quadratic for h. Simplify by multiplying everything by 1000

0 = 249.5 h^2 + 5.994 h - 71.928

use quadratic formula to solve for h

h = 0.525 cm

This is the height of the soda in the can for which the center of mass is at its lowest.

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