Transverse waves on pulley - question

[3psilon]

Homework Statement

1) an aluminium wire, of length L1 = 60.0 cm, cross sectional area 1.00 x 10^-2 cm^2, and density 2.60 g/cm^3, is joined to a steel wire of length L2, of density 7.80 g/cm^3 and the same cross sectional area. The compound wire, loaded with a block of mass m = 10.0 kg, is arranged so that the distance L2 from the joint to the supporting pulley is 86.6 cm. Transverse waves are set up in the wave using an external source of variable

a) find the lowest frequency of excitation for which standing waves are observed such that the joint in the wire is a node
b) How many nodes are observed at this frequency?

Homework Equations

f1= 1/2L * √(T/mu)

The Attempt at a Solution

So far, I know to find tension by basic Ft=Fg
I'm having trouble understanding the theory of it, there will be discontinuity because the change in linear density, but how can I account for that? Also, if there is a node where the lin. density changes, is the discontinuity relevant?

i also tried using percents for my linear density, i.e. 1.466 m = total length of the wire to the pulley, so then 41% of the wire is made up of aluminium, so the linear density could be written as mu = mualuminium + musteel, with adjusted linear density based on the percent

 

[emailanmol]

I am having trouble understanding the description of your system.

How are the strings tied to pulley?

Can you post a picture?

Also is value of L2 given?

 

[emailanmol]

https://www.physicsforums.com/showthread.php?t=216294

Refer to this thread :-)

Absence of picture is what caused the problem there as well.

If you can't post a picture (or a drawing) it would be great

 

[emailanmol]

Ok.I think i understood the problem and have got the answers.(In case you have the final answer let me know so that I can cross check if my answers are right)

there is no discontinuity at the joint.
Its just that at the junction u(linear density)changes, so the velocity changes , similar to cases like refraction of light.

Now,

Since the masses of strings are negligible in comparison to the block you can ignore the change in tension due to weight of strings.
So tension in both strings comes out be nearly mg(m is mass of block).
(the difference is about 10^(-10) which can be ignored)

now what is the ratio of frequency in the two strings?(fairly simple question.)

Now what is ratio of velocities on the two strings?(Remember tension is same.Linear densities are different.)
What are ratio of wavelengths?

(Use v=lambda*f)

If you got these read further.
If you didn't,don't worry.We will focus on the things discussed above first and then move ahead :-)

-------------
Suppose now the no. of loops in string 1 is n and string 2 is k.
What is the relation between lambda1, L1 and n?
Similarly what is the relation for string 2?

For minimum frequency, should n and k be large or small?

 

[asteriatic]

of each material. But I'm still having trouble understanding the theory behind this problem.
Based on the given information, it seems that the setup for this problem involves a compound wire made up of two different materials, aluminium and steel, with different densities. The wire is loaded with a block of mass 10.0 kg and arranged in a way that transverse waves can be set up using an external source.

To find the lowest frequency of excitation for which standing waves are observed with the joint in the wire as a node, we can use the formula f1= 1/2L * √(T/mu), where L is the length of the wire, T is the tension, and mu is the linear density. In this case, the length of the wire (L) is given as 60.0 cm + L2, where L2 is the length of the steel wire. The tension (T) can be found by using Ft=Fg, where Ft is the tension in the wire and Fg is the force due to the weight of the block. However, we also need to take into consideration the change in linear density at the joint in the wire.

Since the linear density is given in units of g/cm^3, we can convert the lengths of the wires from cm to m and the cross-sectional area from cm^2 to m^2. This will give us the linear density in units of kg/m. We can then use the adjusted linear density for the calculation of the lowest frequency of excitation.

As for the number of nodes observed at this frequency, it will depend on the length of the steel wire (L2). Since we are given the distance from the joint to the supporting pulley (86.6 cm), we can use this to find the length of the steel wire (L2) by subtracting it from the total length of the wire (L+L2).

In summary, to solve this problem, we need to take into consideration the discontinuity in linear density at the joint in the wire and use the adjusted linear density for the calculation of the lowest frequency of excitation. Additionally, the number of nodes observed at this frequency will depend on the length of the steel wire.