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        riddles >> medium >> Please Rewind
        
(Message started by: denis on Apr 27^th, 2008, 5:45pm)

Title: Please Rewind
Post by denis on Apr 27^th, 2008, 5:45pm

I got this question on a first year university Physics assignment 25 years ago. I thought it was an interesting question because the answer does not depend on the mass of the tape roll.

Easy-Medium difficulty for a Physics major.... Hard for a non physics students.

--------------------------

A length L of flexible tape is tightly wound. It is then allowed to unwind as it rolls down a steep incline that makes an angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif with the horizontal, the upper end of the tape being tacked down.

Find the time T it takes for the tape to completely unwind.

Title: Re: Please Rewind
Post by william wu on May 20^th, 2008, 3:52pm

I'm not a physics student, but I'll try. My approach is naive and most likely wrong ::) However, I'd be very happy to have someone explain why it's wrong, if it is.

[hideb]
Let's consider any particular instant in time, t. At that moment, the tape roll has mass M(t) and radius R(t); henceforth, for brevity, I will just refer to these as M and R.

Let f denote the friction force the tape roll experiences at that moment.

From the equations F=ma and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/tau.gif= I http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif, we have

(1) Mg sin http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif- f = Ma, where a denotes linear acceleration down the slope.
(2) f R = I a/R, where I is the moment of inertia of the tape roll

We model the tape roll as a solid disk. Thus, I = MR^2, and (2) becomes
(edit: Actually, the moment of inertia should be 1/2 MR^2. See my post later for corrections that account for this error.)

(2) f = M a.

Combining (1) and (2) yields

(3) a = (g/2) sin http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif

an answer which is apparently independent of both M and R.

As time progresses, we will have M(t) > M(t + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif), and R(t) > R(t + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif). But by equation (3) above, that does not affect the acceleration, which is constant. Hence we can use kinematics to finish the problem:

(4) L = (1/2) a t^2

or

t = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{2L/a} = 2 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{ (L/g) csc http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif}.
[/hideb]

Title: Re: Please Rewind
Post by temporary on May 20^th, 2008, 9:00pm

I think you would need the density to know exactly how flexible the tape is.

Title: Re: Please Rewind
Post by denis on May 20^th, 2008, 9:02pm

William,

For someone who is not physics major, you did well indeed. Your approach is correct and well thought out.

You just need a small adjustment: recheck the moment of inertia formula you used for the disk. The formula you used is for a cylindrical shell with open ends which assumes the shell thickness is negligible (which is not the case for the solid disk).

Title: Re: Please Rewind
Post by denis on May 20^th, 2008, 9:04pm

on 05/20/08 at 21:00:35, temporary wrote:

I think you would need the density to know exactly how flexible the tape is.

You can assume no friction or resistance arises from the tape unrolling other than inertial forces.

Title: Re: Please Rewind
Post by william wu on May 20^th, 2008, 10:15pm

Thanks denis!

Scaling factor corrections:
- I = 1/2 MR^2, and thus a = 2/3 g sin http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif
- t = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{ 2L/a } = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{ (3L/g) csc http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif}

Title: Re: Please Rewind
Post by temporary on May 20^th, 2008, 10:26pm

on 05/20/08 at 21:04:10, denis wrote:

You can assume no friction or resistance arises from the tape unrolling other than inertial forces.

But that doesn't give me the density.

Title: Re: Please Rewind
Post by ThudanBlunder on May 21^st, 2008, 3:48am

on 05/20/08 at 22:26:14, temporary wrote:

But that doesn't give me the density.

You must be a member of DENSA.

Title: Re: Please Rewind
Post by denis on May 21^st, 2008, 8:40am

on 05/20/08 at 22:15:07, william wu wrote:

Yep!