Solve Rope Physics Problem: Find Time to Travel Length of Rope

[DMD]

Here's the problem:

A uniform rope with length L and mass m is held at one end and whirled in a horizontal circle with angular velocity omega. You can ignore the force of gravity on the rope. Find the time required for a transverse wave to travel from one end of the rope to the other.




So far by fiddling with some equations, I've got:

v = sqrt(F_tension/µ)

F_tension = ma = m*L*omega^2

µ = m/L

v = sqrt((m*L*omega^2)/(m/L)) = sqrt(omega^2*L^2) = omega*L


And of course, that gets me nowhere except another textbook equation, namely v = R*omega

So I have no idea where to go from there.

Any help will be appreciated.

 

[andye]

Your equations look a bit fishy. For example, your last relation v=L*omega implies that the units of angular velocity are just meters, which isn't correct. Maybe you can relate L (angular momentum) in terms of distance L and linear density lambda and omega (or one of its derivatives?) by using the moment of inertia. Once we know the velocity, we already have the length, so the time it takes for a wave to travel down the rope's length should be easy. g'luck!

 

[wais]

To solve this problem, we can use the formula for the speed of a wave, v = sqrt(F_tension/µ), where F_tension is the tension in the rope and µ is the linear density of the rope. We can also use the fact that the speed of a wave is equal to the wavelength divided by the period, v = λ/T.

First, let's find the tension in the rope. We know that the tension in the rope is equal to the centripetal force, F_c = m*v^2/R, where m is the mass of the rope, v is the speed of the rope, and R is the radius of the circle. Since the rope is being whirled in a horizontal circle, the centripetal force is provided by the tension in the rope, so F_c = F_tension. Therefore, F_tension = m*v^2/R.

Next, we need to find the linear density of the rope, µ. This can be calculated by dividing the mass of the rope by its length, µ = m/L.

Now, we can substitute these values into the equation for the speed of the wave, v = sqrt(F_tension/µ). This gives us:

v = sqrt((m*v^2/R)/(m/L))

Simplifying, we get:

v = sqrt(v^2*R/L)

Finally, we can use the equation for the speed of a wave, v = λ/T, and substitute in the speed we just found, v = sqrt(v^2*R/L), to solve for the period, T:

v = λ/T

sqrt(v^2*R/L) = λ/T

T = λ/sqrt(v^2*R/L)

Since the wavelength, λ, is equal to the length of the rope, L, we can substitute that in:

T = L/sqrt(v^2*R/L)

Simplifying, we get:

T = L*sqrt(L/R)

Therefore, the time required for a transverse wave to travel from one end of the rope to the other is T = L*sqrt(L/R). This is the final solution to the problem.