Question

In a uniform, freely hanging rope wave velocity v varies with the distance z from the top. This is because the tension T varies while the mass per unit length μ remains constant. Suppose you want to design a freely hanging rope in which wave velocity v is independent of z. For such a rope, μ cannot be constant, and you must specify the function μ(z).

The desired form of μ(z) if the rope is infinitely long is [μ₀ is the mass per unit length of the rope at the top (z = 0)] :-

• A. μ(z) = μ₀e^(-(g/v²)z)
• B. μ(z) = μ₀[1 - (g/v²)z]
• C. μ(z) = μ₀e^(-(2g/v²)z)
• D. μ(z) = μ₀[1 + (g/v²)z]

Answer 1

Answer: (A)

μ(z) = μ₀e^(-(g/v²)z)

Answer 2

For a freely hanging rope with constant wave velocity v, the mass per unit length μ(z) at a distance z from the top satisfies the differential equation v² dμ/dz = -μ(z) g.

Integrating this with the boundary condition μ(0) = μ₀ gives μ(z) = μ₀ e^(-(g/v²) z).