How Do You Solve an Oscillating Rope Problem?

[tigers4]

Homework Statement

A rope, under tension of 120 N and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by
y = (1.18 m)sin ( x / 2) sin (10 t),
where x = 0 at one end of the rope, x is in meters, and t is in seconds.

(a) What is the length of the rope?

(b) What is the speed of the waves on the rope?

(c) What is 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

v=sqrt(t/density)
d=m/v

The Attempt at a Solution

I have no clue how to start this problem, can anyone provide a way to approach a problem like this?

 

[vela]

What does the second harmonic look like? Specifically, how many nodes are there and where are they located on the string?

 

[tigers4]

the second harmonic has three nodes

 

[vela]

Right, there are nodes at the ends of the string and one right in the middle. Now try plotting y vs. x and seeing where its zeros are. Those zeros correspond to the nodes on the string.

 

[rwisz]

I would approach this problem by first identifying the given information and understanding the concept of standing waves. I would also refer to any relevant equations, such as the wave equation and the relationship between wave speed, tension, and density.

(a) To find the length of the rope, I would use the equation for the wavelength of a second-harmonic standing wave:
λ = 2L, where λ is the wavelength and L is the length of the rope. Rearranging the equation, we get L = λ/2.
The wavelength can be found by using the formula λ = 2π/k, where k is the wave number. In this case, k = 2π/2 = π.
Therefore, the length of the rope is L = (2π/π)/2 = 1m.

(b) The speed of the waves on the rope can be calculated using the equation v = √(T/μ), where T is the tension and μ is the linear mass density of the rope. Rearranging the equation, we get μ = T/v².
Substituting the given values, we get μ = 120 N / (1.18 m)² = 79.66 kg/m.
Using the equation for the speed of a wave, v = √(T/μ), we get v = √(120 N/79.66 kg/m) = 3.76 m/s.

(c) The mass of the rope can be calculated by dividing the linear mass density by the length of the rope. Therefore, the mass of the rope is m = 79.66 kg/m / 1m = 79.66 kg.

(d) If the rope oscillates in a third-harmonic standing wave pattern, the period of oscillation can be found using the formula T = 2π/ω, where ω is the angular frequency. In a third-harmonic standing wave, there are three antinodes between two adjacent nodes. Therefore, the wave number k = 3π.
Using the relationship between angular frequency and wave number, ω = v*k, where v is the speed of the wave. Substituting the values, we get ω = (3.76 m/s)*(3π) = 11.28 π rad/s.
Finally, the period of oscillation is T =