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Question

A string, fixed at both ends, vibrates in a resonant node with a separation of 2.0 cm between the consecutive nodes. For the next higher resonant frequency, this separation is reduced to 1.6 cm. find the length of the string.


A

18 cm

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B

8 cm

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C

10 cm

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D

cannot be determined

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Solution

The correct option is B

8 cm


In the first case the distance between cause cu tine nodes is 2 cm,

Which implies λ2 = 2cm cm or λ = 4cm

If v is the velocity of wave on the string

Frequency,f = f = vλ = v4- - - - -- (1)

When the distance between the nodes become 1.6 cm

λ2 = 1.6cm λ=3.2cm

Frequency,f = vλ = v3.2 - - - - - (2)

Also for some n,f = nv2L

v4 = nv2L [from(1)]

n = L2 - - - - - (3)

then for some n+1,f = (n+1)v2L

v3.2 = (n+1)v2L [from(2)]

2L3.2 = n + 1 ----------- (4)

From (3) and (4)

2L3.2 = L2 + 1

4L = 3.2 (L+2)

4L = 3.2L + 6.4

0.8L = 6.4

L = 8cm


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