Question

Two children stretch a jump rope between them and send wave pulses back and forth on it. The rope is 2.7 m long, its mass is 0.46 kg, and the force exerted on it by the children is 37 N.

(a)
What is the linear mass density of the rope (in kg/m)?



(b)
What is the speed of the waves on the rope (in m/s)?

Answer 1

(a) The linear mass density ρ of the rope is its mass m per unit of length:

$$\rho =\frac{m}{L}$$

Substituting m=0.46 kg and L=2.7 m we get

$$\rho =\frac{0.46\,kg}{2.7\,m}\approx 0.17\frac{kg}{m}$$

So the linear mass density of the rope is 0.17 kg/m

(b) If the tension in the rope is T, and the linear density is ρ, then the speed of the waves u on the rope is determined by

$$u=\sqrt{\frac{T}{\rho }}$$

Substituting T=37 N and ρ=0.17 kg/m we get

$$u=\sqrt{\frac{37\,N}{0.17\,\frac{kg}{m}}}=\sqrt{\frac{37\,kg\cdot \frac{m}{{{s}^{2}}}\,}{0.17\frac{kg}{m}\,}}\approx 14.8\,\frac{m}{s}$$

So the speed of the waves on the rope is 14.8 m/s