Standing Wave Problem: Aluminum and Steel Wire Joint with External Source

[ductape]

an aluminum wire of length L1 = 50.0 cm, cross-sectional area 1.00×10-2 cm2, and density 2.70 g/cm3, is joined to a steel wire of density 7.80 g/cm3 and the same cross-sectional area. The compound wire, loaded with a block of m = 10.0 kg, is arranged so that the distance L2 from the joint to the supporting pulley is 44.13 cm. Transverse waves are set up in the wire by using an external source of variable frequency; a node is located at the pulley. Find the lowest frequency of excitation for which standing waves are observed such that the joint in the wire is one of the nodes.

Having some difficulty with this problem. First I calculated the mass and linear density of each section of the rope. Using v^2 = T/u the wavespeed on each section was calculated. However, this gives a different frequency for each section, so I'm obviously doing something wrong. Any suggestions?

 

[ductape]

haha nevermind got that one literally 1 min after posting...just came to me.

 

[m2003]

To solve this problem, we need to consider the properties of standing waves and how they relate to the given parameters. A standing wave is a wave that appears to be standing still, but is actually the result of two waves with the same frequency and amplitude traveling in opposite directions. In order for standing waves to occur, the distance between the nodes (points of no displacement) must be an integer multiple of half the wavelength.

In this case, we have a joint in the wire, which means that it cannot move and must be a node. This means that the distance between the joint and the pulley (L2) must be an integer multiple of half the wavelength. So, we can set up an equation using the wavelength formula, λ = 2L/n, where n is the number of nodes.

Next, we need to consider the frequencies of the waves on both the aluminum and steel sections of the wire. As you correctly calculated, the wavespeed on each section is different due to the difference in density. This means that the frequency of the waves on each section will also be different. However, in order for standing waves to occur, the frequencies on both sections must be the same.

To find the lowest frequency of excitation for which standing waves are observed, we need to find the lowest frequency that satisfies both the wavelength equation and the frequency equation. This can be done by setting the two equations equal to each other and solving for the frequency.

Once we have the lowest frequency, we can use the equation f = v/λ to find the corresponding wavelength. This wavelength will then determine the number of nodes and the distance between them, allowing us to confirm that the joint is indeed one of the nodes.

In summary, to solve this problem, we need to consider the properties of standing waves and how they relate to the given parameters. We also need to ensure that the frequencies on both sections of the wire are the same in order for standing waves to occur. By setting up and solving equations, we can determine the lowest frequency of excitation for which standing waves are observed, and confirm that the joint is one of the nodes.