How Long Does It Take for 10 Meters of a Rolled Flag to Unwrap?

[Misheel]

Homework Statement

The flag is d=2mm thick and D=30 m long. We roll the flag and stick the short side of the flag (long side is 30m, short side's length is not given) on the ceiling so that the flag unwraps back to the ground. Flag is M=30kg.
1. Find the time it takes to 10 m of the flag to get unwrapped (meaning 20 m is still in cylinder mode)
2. Find the acceleration at that given point in time.

p.s sorry for my bad English

Homework Equations

i guess :
D/M=p (linear density)

i am really not sure where to approach this problem.

The Attempt at a Solution

i assumed that the rolled flag is just like the piled up cylinders each ones radius is d=2mm lesser than the outside one. and assumed the most inner cylinder had r0=2mm radius. So to find the most outer radius of the rolled flag i used the arithmetic progress:
2*pi* (r0+r1+r2+r3+...rn)=30m
(r0+rn)/2*n=15/pi
n=(approx) 22.31 which has to be integer so -> 22 (?)
and the most outer radius wud be rn=44*10^(-3)m=4.4cm

and after this i have no clue where to approach this problem
mass is changing, radius is changing... don't know where to pick the point if to write a torque equation :P
any advice would be really helpful. Thank you

 

[ehild]

Hi Misheel, welcome to PF!

It is not an easy problem. I would use conservation of energy in terms of the wrapped off length. Note that the flag is rolling down: apply the rolling condition.
It is also needed to find the relation between the length of flag still wrapped on and the radius of the cylinder. Your approach with the arithmetic progression is a good start, but do it for radius and length in general.

ehild

 

[Misheel]

I would use conservation of energy in terms of the wrapped off length.

so... so it means i have to find the relation between rotating flag's potential and rotating energy, and wrapped off flag's potential energy ? and the mass is changing right...
also

relation between the length of flag still wrapped on and the radius of the cylinder.

one is what i can't find >.< ...

 

[Orodruin]

Try instead to find a relationship between the length of flag left in the cylinder and the radius. Hint: As a first crude approximation, compare flag volume with cylinder volume.

 

[ehild]

so... so it means i have to find the relation between rotating flag's potential and rotating energy, and wrapped off flag's potential energy ? and the mass is changing right...

The whole flag has potential energy which is the sum of the wrapped-off part and part till wrapped on. The kinetic energy consist of that of translational motion of the CM and rotation about the axis of the cylinder.

You have figured out that the radii of the layers make arithmetic progression. So the radius of the cylinder when there are n layers on it is r_n= nD (D=0.002 m) and the length of the flag L=2pi(r_n/2)n. Eliminate n and you get the function L(r)

 

[Misheel]

sorry for asking a lot >.< but
translational motion is the motion of the object consisted of both straight and rolled part right ? (which means CM is in the air if we picture it ?)
and then where is the time coming ?

and thanks for all the advice :D

 

[ehild]

Yes, the CM of the whole system translates downward, but the system consist of the cylindrical part which moves downward and the hanging piece of the flag which is in rest.

If you choose the energy approach, you might get the velocity as v=dL/dt as function of L, the unwrapped length. You can get the time by integration.