Wave Speed Question Homework: Length, Speed & Mass

[blindzero678]

Homework Statement

A rope, under a tension of 200 N and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by

y = (0.10m)(sinPIx/2)sin12PIt

where x = 0 at one end of the rope, x is in meters, and t is in seconds. What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c) the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

Homework Equations

I'm not sure...I know how to solve for the length of the rope, but I don't know which formula to use to solve for the wave speed. Although once i find this, I know how to find the mass of the rope and the period in part (d).

The Attempt at a Solution

I know you can find the length of the rope by setting

0.10 = 0.10sin(PIx/2)

because the 0.10 m in the original equation will be the amplitude of the wave, and at it's maximum, sin(12PIt) = 1. this gives you x = 1 meter.

But i don't know where to go from here, to get the velocity.

i know f = v/lambda = (nv)/(2L) and that lambda = 2L/n, but you can't use these equations to solve for velocity, because the v cancels out.

 

[Redbelly98]

There is a problem with (a). The answer does not depend on the amplitude.

Have you drawn a picture of the rope for the second-harmonic pattern? Do that, and think about how wavelength and rope length are related.

 

[thomate1]

I would first clarify the question and make sure that the units for all the given values are consistent. In this case, the tension is given in newtons, while the displacement is given in meters. I would recommend converting the tension to SI units (kilograms and meters per second squared) to make the calculations more accurate.

To solve for the wave speed, I would use the equation v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. In this case, the frequency can be determined by looking at the argument of the sine function, which is 12πt. This means that the frequency is 12π cycles per second, or hertz. To find the wavelength, we can use the fact that the second-harmonic standing wave has two nodes (points of zero displacement) between the two fixed ends. This means that the wavelength is twice the length of the rope, or 2 meters.

Plugging in the values, we get v = (12π Hz)(2 m) = 24π m/s. This is the speed of the waves on the rope.

To find the mass of the rope, we can use the equation ρ = m/L, where ρ is the linear density (mass per unit length) of the rope, m is the mass, and L is the length. We already know the length of the rope (1 meter), so we just need to find the linear density. To do this, we can use the equation ρ = T/μv^2, where T is the tension, μ is the linear mass density (mass per unit length) of the rope, and v is the wave speed. Plugging in the values, we get ρ = (200 N)/(24π m/s)^2 = 0.0026 kg/m. Multiplying this by the length of the rope, we get a mass of 0.0026 kg.

For part (d), to find the period of oscillation for a third-harmonic standing wave pattern, we can use the equation T = 1/f, where T is the period and f is the frequency. In this case, the frequency is 3 times the frequency of the second-harmonic wave, so the period is 1/3 of the period of the second-harmonic wave. Therefore, the period for the third-harmonic wave is (