Question 183: Waves

Question

A wire of length 2L, is made by joining two
wires A and B of same length but different radii
r and 2r and made of the same material. It is
vibrating at a frequency such that the joint of
the two wires forms a node. If the number of
antinodes in wire A is p and that in B is q then
the ratio p : q is :

Accepted Answer

Let mass per unit length of wires are $$\mu $$
1
and $$\mu $$
2
respectively
$$ \because $$  Materials are same, so density $$ho $$ will be same.
$$ 	herefore $$ $${\mu _1} = {{ho \pi {r^2}L} \over L} = \mu $$ and $${\mu _2} = {{ho 4\pi {r^2}L} \over L} = 4\mu $$
Tension in both are same = T, let speed of wave in wires are V
1
and V
2
$${V_1} = {{{V_1}} \over {2L}} = {V \over {2L}}\,\,\& \,{V_2} = {{{V_2}} \over {2L}} = {V \over {4L}}$$
Frequency at which both resonate is L.C.M. of
both frequencies (i.e : $${V \over {2L}}$$ )
Hence number of loops in wires are 1 and 2 respectively
So, ratio of number of antinodes is 1 : 2.

Answer

3 : 5

Answer

4 : 9

Answer

1 : 2

Answer

1 : 4

Waves

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