Question

A transverse harmonic wave travels on a rope according to the following expression: y(x,t) = 0.17sin(2.8x + 17.3t) The mass density of the rope is μ = 0.107 kg/m. x and y are measured in meters and t in seconds.

1)

What is the tension in the rope?

2)

At x = 3.5 m and t = 0.4 s, what is the velocity of the rope? (watch your sign)

3)

At x = 3.5 m and t = 0.4 s, what is the acceleration of the rope? (watch your sign)

4)

What is the average speed of the rope during one complete oscillation of the rope?

5)

In what direction is the wave traveling?

6)

On the same rope, how would increasing the wavelength of the wave change the period of oscillation?

Answer 1

Since you didn't list question 1...
2: I will solve in a minute.
3: Looking at the equations we see k=(2pi)/wavelength. Since k is given in the problem you can solve easily for wavelength.
Now that you know wavelength you can use the equation omega/k=frequency*wavelength. Omega is given in the problem already so now you can solve for frequency.
4. Again, just by looking at the equations, we see that Vwave = omega/k = frequency*wavelength.
5. Tension in the rope is solved by using the equation Vwave = sqrt(T/?). ? is given in the problem, so just plug in your values.
6. You have to differentiate 0.14sin(2.2x + 17.8t) with respect to time, which becomes 0.14*17.8*cos(2.2x+17.8t). Just plug in your values for t and x, and make sure your calculator is in radians.
7. Now you have to differentiate the velocity equation, which gives you -0.14*17.8*17.8sin(2.2x+17.8t).
8. Average speed is, as far as I know, (2*Amplitude*omega)/pi.
9. Now I don't know if I got the answer right, but my professor said that when the values of kx and omega*t have the same sign, the wave moves to the left, and when they are opposite signs the wave moves to the right. So in this case it would move left/ in the -x direction.
10. Again, I don't know if this is 100% right, but we know that with a wave V=wavelength*frequency, so if wavelength goes up, frequency has to go down because velocity stays the same (like how the speed of light is constant). So since frequency went down, we know that f=1/T where T is the period. So you can see that when f goes down, T goes up. So the period would increase.

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