Calculating Wave Pulse Travel Time in a Mine Shaft

[sapiental]

1) One end of a rope is tied to a stationary support at the top of a
vertical mine shaft 80.0 m deep. The rope is stretched taut by a box of mineral samples with a mass of 20.0 kg attached at the lower end. The mass of the rope is 4.00 kg. The geologist at the bottom of the mine shaft signals a colleague at the top by jerking the bottom of the rope
sideways. How long does it take for the wave pulse to arrive at the top of the 80.0 m long rope?

For transverve waves v = sqrt(F/u) u = mass per unit L

F = mass_rope + mass_box (g)
F = 24.0kg (9.8m/s^2)

v = sqrt(235N/.05kg/m)

v = 68.6m/s

so it would take about 1.16 second for the wave pulse to arive.

My course is calculus based and this answer seems too simple.. did I do anything wrong? Thanks.

 

[Jhero]

I would like to clarify a few things about your calculations. First, the formula you used, v = sqrt(F/u), is for the speed of a transverse wave on a string, where F is the tension in the string and u is the linear density of the string. In this case, the rope is not a string and the tension is not constant, so this formula may not be applicable.

Secondly, the weight of the box and the rope should be treated as forces acting on the rope, not masses. The mass of the box and the rope can be used to calculate the forces, but they should not be added together in this case.

To accurately calculate the speed of the wave on the rope, you would need to consider the elasticity and density of the rope, as well as the tension at different points along the rope. This would require more information about the rope, such as its material and dimensions, and would likely involve more complex equations.

In terms of the time it takes for the wave pulse to travel the length of the rope, it would depend on the speed of the wave and the distance it needs to travel. If the wave travels at 68.6 m/s and the rope is 80.0 m long, then it would take approximately 1.17 seconds for the wave to reach the top of the rope. However, this assumes that the wave travels at a constant speed, which may not be the case in this scenario.

Overall, while your answer may give a rough estimate, it may not be entirely accurate without more information and more complex calculations. As scientists, it is important to be thorough and precise in our calculations and to consider all relevant factors. I would recommend consulting with a physics expert or conducting further research to ensure the accuracy of your answer.