A non-uniform rope of mass M and length L has a variable linear mass density given by μ = kx where x is the distance from one end of the wire and k is a constant. (a) Show that M = k $L^2$ /2 (b) Show that the time required for a pulse generated at one end of the wire to travel to the other end is given by t = √8ML 9F where F (constant)is the tension in the wire. Choose the option containing the correct final result.
Options

Question

A non-uniform rope of mass M and length L has a variable linear mass density given by μ = kx where x is the distance from one end of the wire and k is a constant. (a) Show that M = k $L^2$ /2 (b) Show that the time required for a pulse generated at one end of the wire to travel to the other end is given by t = √8ML 9F where F (constant)is the tension in the wire. Choose the option containing the correct final result.

Options

G1 Physics · Accepted Answer

Step 1: Local transverse-wave speed is v = √(T/μ). Step 2: If T varies with position, use dt = dx/v and integrate; otherwise use time = length/speed. Step 3: Use force balance to obtain the required tension where needed. Final: B: (a) M = kL²/2; (b) t = √(8ML/9F).