Question

A steel cable has a cross-sectional area 4.49 × 10^-3 m^2 and is kept under a tension of 2.96 × 10^4 N. The density of steel is 7860 kg/m^3. Note that this value is not the linear density of the cable. At what speed does a transverse wave move along the cable?

Answer 1

The speed of a transverse wave along the steel cable is 122.47 m/s.

Step-by-step explanation:

The speed of a transverse wave along a steel cable can be calculated using the formula:

Speed of wave = √(tension/linear mass density)

First, we need to calculate the linear mass density of the cable. The linear mass density is the mass per unit length of the cable. We can find it using the formula:

Linear mass density = density x cross-sectional area

Using the given values, the linear mass density of the cable is 7860 x 4.49 x 10^-3 = 35.28 kg/m.

Plugging this value into the formula, we get:

Speed of wave = √(2.96 x 10^4 / 35.28) = 122.47 m/s

Answer 2

The transverse wave will travel with a speed of 25.5 m/s along the cable.

Step-by-step explanation:

let T = 2.96×10^4 N be the tension in in the steel cable, ρ = 7860 kg/m^3 is the density of the steel and A = 4.49×10^-3 m^2 be the cross-sectional area of the cable.

then, if V is the volume of the cable:

ρ = m/V

m = ρ×V

but V = A×L , where L is the length of the cable.

m = ρ×(A×L)

m/L = ρ×A

then the speed of the wave in the cable is given by:

v = √(T×L/m)

= √(T/A×ρ)

= √[2.96×10^4/(4.49×10^-3×7860)]

= 25.5 m/s

Therefore, the transverse wave will travel with a speed of 25.5 m/s along the cable.