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

Answer 1

Answer:

Let mass per unit length of wires are μ

1

and μ

2

respectively.

∵ Materials are same, so density ρ is same.

∴μ

1

=

L

ρπr

2

L

=μ and μ

2

=

L

ρ4πr

2

L

=4μ

Tension in both are same =T, let speed of wave in wires are V

1

and V

2

V

1

=

μ

T

=V.V

2

=

4μ

T

=

2

V

So fundamental frequencies in both wires are f

01

=

2L

V

1

=

2L

V

and f

02

=

2L

V

2

=

4L

V

Frequency at which both resonate is L.C.M of both frequencies i.e.

2L

V

.

Hence no. of loops in wires are 1 and 2 respectively.

So, ratio of no. of antinodes is 1:2.

solution