Question

For this assignment, you should mathematically solve and record a video testing your solution for the following prompt: Two rolls of toilet paper, of equal mass and radius, are dropped from different heights so that they hit the ground at the same time. One roll of toilet paper is dropped normally while the other is dropped while a person holds onto a sheet of toilet paper such that the roll unravels as it descends. Determine the ratio of heights h1/h2, where h1 represents the height of the toilet paper dropped normally and h2 represents the height of the toilet paper that unravels, so that both rolls hit the ground at the same time.

Answer 1

Answer:

h1/h2 = $\frac{2R^2}{3R^2 + h^2}$

Explanation:

Using two rolls of tissue paper : One roll dropped normally while the other drops as some holds onto a sheet of the toilet paper ( I.e. the tissue paper drops rotating about its axis )

Determine the ratio of heights  h1/h2

mass of tissues = same

radius of tissues = same

h1 = height of tissue 1

h2 = height of tissue 2

For the first tissue ( Tissue that dropped manually )

potential energy = kinetic energy

mgh = 1/2 mv^2  

therefore the final velocity ( v^2 ) = 2gH  ----- ( 1 )

second tissue ( Tissue that dropped while rotating )

gh = $\frac{v^2}{u}$ ( 3 + $\frac{u^2}{R^2}$ ) ------ ( 2 )

To determine the ratio of heights we will equate equations 1 and 2

hence :

gh = $\frac{2gH}{u}$ ( 3 + $\frac{u^2}{R^2}$ )

∴ h1/h2 = $\frac{2R^2}{3R^2 + h^2}$