In this example we will apply the equation v = F T to the simple case in which t

Question

In this example we will apply the equation v=FT to the simple case in which the tension in a rope is provided by the weight of a box. One end of a nylon rope is tied to a stationary support at the top of a vertical mine shaft that is 80.0 m deep (Figure 1) . The rope is stretched taut by a box of mineral samples with mass 20.0 kg suspended from the lower end. The mass of the rope is 2.00 kg. The geologist at the bottom of the mine signals to his colleague at the top by jerking the rope sideways. What is the speed of a transverse wave on the rope? If a point on the rope is given a transverse simple harmonic motion with a frequency of 20 Hz, what is the wavelength of the wave?

SET UP We can calculate the wave speed v from the tension, FT, and the mass per unit length, =m/L. The tension at the bottom of the rope is equal to the weight of the 20.0 kg load:

FT==(20.0kg)(9.80m/s2)196N

and the mass per unit length is

=mL=2.00kg80.0m=0.0250kg/m

SOLVE The wave speed is given by the following equation:

v=FT=196N0.0250kg/m=88.5m/s

From the following equation,

=vf=88.5m/s20.0s1=4.43m

REFLECT We have ignored the 10% increase in tension in the rope between bottom and top due to the rope's own weight. Can you verify that the wave speed at the top is 92.9 m/s?

Part A - Practice Problem:

Another box of samples is hoisted up by the same rope. If the rope is shaken with the same frequency as before, and the wavelength is found to be 6.6 m , what is the mass of this box of samples?

Express your answer in kilograms to three significant figures.

Explanation / Answer

lambda =v/f

v = f*lambda

v = 20 * 6.6 = 132 m/s

v = sqrt(T/mu)

v = sqrt(mg/0.0250)

v^2 = mg/0.0250

m = v^2*0.0250/g

m = 44.4kg

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