Question

A 1.50-m-long wire having a mass of 0.100 kg is fixed at both ends. The tension in the wire is maintained at 25.0 N.

(a) What are the frequencies of the first three allowed modes of vibration?
f₁ = 6.4549 Hz
f₂ = 12.9099 Hz
f₃ = 19.3649 Hz

(b) If a node is observed at a point 0.300 m from one end, in what mode and with what frequency is it vibrating? (Select all that apply.)

The frequency is the tenth state at 64.5 Hz

The frequency is the second state at 12.9 Hz

The frequency is the fifth state at 32.3 Hz

The frequency is the twentieth state at 129.1 Hz

The frequency is the fifth state at 16.1 Hz

The frequency is the tenth state at 12.9 Hz

*PART A IS CORRECT* I am having trouble with part B, I've tried selecting options:

1, 2, and 3 ;    just 3 ; 2 and 3 ; 1, 2, 3, and 4

and all of them have been incorrect

Answer 1

I am helping with part B only (assuming part A is clear to you.)

Note that nodes are found at x = kλ/2, where k is a positive integer.

Also, λ = 2L/n for string fixed at both ends (Here, L is length of string).

Let's check each frequency one by one.

1) f10 = 10*f1 = 10*6.4549 = 64.5 Hz.

Also, λ = 2L/n = 2*1.5/10 = 0.3 m.

For k = 2, Node position x = 2*λ/2 = 0.3 m. Hence, node exists at 0.3 m in this case.

2) f2 = 2*f1 = 2*6.4549 = 12.9 Hz.

  Also, λ = 2L/2 = 1.5 m

For nodes, x = 0, λ/2, λ = 0 m, 0.75 m, 1.50 m. Hence, node does not exist at 0.3 m in this case.

3) f5 = 5*f1 = 5*6.4549 = 32.3 Hz.

   Also, λ = 2L/5 = 2*1.5/5 = 0.6 m

For nodes, x = 0, λ/2, λ, 3λ/2, 2λ, 5λ/2 = 0 m, 0.3 m, 0.6 m, 0.9 m, 1.2 m, 1.5 m. Hence, node exists at 0.3 m in this case.

4) f20 = 20*f1 = 20*6.4549 = 129.1 Hz.

   Also, λ = 2L/20 = 2*1.5/20 = 0.15 m

For k = 4, Node at x = kλ/2 = 4λ/2 = 2λ = 2*.15 = 0.3 m. Hence, node exists at 0.3 m in this case.

We need not check the last two since the frequency of 5th and 10th state is 32.3 Hz and 64.5 Hz respectively.

Hence, options 1, 3 and 4 are correct.