Tension of a cable as a result of a pulse (wave)?

[LastXdeth]

Homework Statement

A ski gondola is connected to the top of a hill by a steel cable of length 620 m and diamter 1.5 cm. As the gondola comes to the end of its run, it bumps into the terminal and sends a wave pulse along the cable. It is observed that it took 16 s for the pulse to return.

(a) What is the speed of the pulse
(b)What is the tension in the cable?

Homework Equations

simple velocity equation: v = d/t
density equation: p = m/v
speed of wave on a cord: v = √[(F)/(m/L)]

The Attempt at a Solution

I was sucessful with part a of the question. It's just a simple velocity equation: v = 620/16 = 38.75 s.

For the second part, I know I need to find mass, so I could plug it in the speed of wave in a cable equation. I tried to use the simple density equation since I already have information for the cross-sectional part of the cable:
m = pv
m= p (LA)→length times area
m = ?

It seems I don't have enough information to find mass because I don't have p (density)!

 

[RTW69]

You can find the density of steel online. It ranges from 77500 to 80500 kg/m3 depending on alloy

 

[Redbelly98]

It seems I don't have enough information to find mass because I don't have p (density)!

Many physics textbooks have a table of densities for different materials. It would be in the section of the chapter that discusses density. They probably expect you to use the value from such a table -- rather than finding it on the web, which would have some variability since there are actually different types of steel with different densities.

 

[LastXdeth]

Thanks, it never came across my mind that density was a given value! I will check my textbook.

 

[asteriatic]

You are correct that you do not have enough information to find the mass or density of the cable. However, you can use the information given to calculate the tension in the cable using the speed of wave equation. In this case, the mass of the cable is not necessary.

To find the tension, you can rearrange the equation to solve for F (tension):

F = (m/L)v^2

Since you do not have the mass or length, you can instead use the linear density (mass per unit length) of the cable, which is given by:

p = m/L

Rearranging this equation, you can solve for m/L:

m/L = p

Now, you can substitute this into the equation for tension:

F = pv^2

Plugging in the values given in the problem, you get:

F = (0.015 kg/m)(38.75 m/s)^2

Solving for F, you get a tension of approximately 22,404 N.

Note: It is important to mention that this calculation assumes that the wave pulse travels at the same speed as the gondola, which may not be entirely accurate. However, it provides a good estimate for the tension in the cable.