dt = dy/√( (LFt) / (M/L) )

Question

Hello, I am doing some physics HW and I am having trouble with a problem. this is what I know so far. v = dy/dt, but I am trying to solve for time (t) so the equation becomes dt = dy/v, and v = ( (LFt) / (M/L) ). so I need to solve dt = dy/√( (LFt) / (M/L) ).

note: I think Ft is tension force in this case.

The question:

A long rope of mass M and length L is suspended vertically. If a student gives a the bottom of the rope a quick shake, how long does it take the pulse to travel up the entire length of the string?

t = ?

I can solve t in terms of M, L, g, pi, and constants.

thank you for the help.

Answer 1

Hi Mehmed,

Quick clarification question: is there supposed to be a square root around v? I noticed that you put a parenthesis outside the v in "v = ( (LFt) / (M/L) )", and there is a square root around it in the expression "dt = dy/√( (LFt) / (M/L) ). ".

If that's the case, and assuming everything you have provided is correct (there might be a minor mistake in v, see the end of the answer), let's proceed as follows (if not, feel free to change it to the expression without the square root)

1.Simplify the differential equation to make your life easier :)

If the 2 Ls are the same variable in v, we can simplify it as

v = L (√ F_t) /(√M)

Then, the differential equation, as you mentioned, becomes

dt = [ (√M) / L (√ F_t)] * dy

2.Solving the Differential Equation to get t

To find the total time t, we want to integrate both sides from 0 to L, which is the entire length of the rope because this is the length that it needs to travel

t = ∫^L₀ [ (√M) / L (√ F_t)] * dy, from 0 to L (apologies, I don't know how to format on this website properly)

Since [ (√M) / L (√ F_t)] is constant with respect to y, we can take it outside the integral,

t = [ (√M) / L (√ F_t)] * ∫^L₀ dy, from 0 to L

Then, solving the integral, we have

t = [ (√M) / L (√ F_t)] * y |^L ₀ = [ (√M) / L (√ F_t)] * (L - 0) = [ (√M) / L (√ F_t)] * L

Notice that the L can cancel out, and now we have

t = (√M) / (√ F_t)

3.Find F_t to answer the question (Wahoo!)

Now we have t as an expression, but we have a problem: Ft is not permitted as a variable! To resolve this, we want to express Ft in a permitted variable.

For a vertically hanging rope, the tension at any point is due to the weight of the rope below that point. You can verify this by drawing a force diagram. Therefore, at the very top of the rope at length L, the tension F_t is the weight of the entire rope becomes

F_t = M*g

Substitute F_t = M g into the equation we have at the end of part 2, we have

t = (√M) / (√ F_t) = (√M) / (√ M * g) = 1/√g

Potential Mistake?

However, I have to point out that usually, the wave speed v is not constant along the length of the rope because tension T varies with height y (as I previously mentioned). To fix this, simply write

v(y) = √[T(y) / μ]

where μ is the linear density M/L, which yields

v = √ [(M/L) * g * y / (M/L)] = √gy

Repeat the approach I have given above, we should arrive at the answer t = 2√[L/g]

If you believe this was a possible mistake that you have made, feel free to reach out to me, and I can guide you through the rest of the derivation

Thank you! I hope this is helpful, and have a great day :)

Devany