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The fundamental frequency of sonometer wire is n. If the length, tension and diameter are tripled, the new fundamental frequency is:
A.   n3B.   n3C.   n3D.   n33

Answer
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Hint: First calculate the frequency of sonometer wire at its original length, diameter and tension. After that calculate the frequency of sonometer wire when length, diameter, and tension is tripled. Compare both the frequency to get a relation between them.

Formula used:
Frequency of oscillation of any wire of length l mass m and tension on string T is
f=12lTm
Mass of any material of volume Vand density ρis m=Vρ
VolumeV= length× area of cross-section
Area of cross-section of a surface of diameter d is A=πd24

Complete answer:
Let the initial length of the wire be l,diameter be d and the tension on the wire be T, then the frequency is given by
n=12lTm
If the wire has density=ρ. Then mass of the object is given by

m= volume×densitym=Area×length×densitym=πd24×l×ρ=πd2lρ4

 So, n=12lT(πd2lρ4)

When the length, diameter and tension is tripled then the new frequency will be
n=12lTm
Here l=3l,T=3T,
The mass of the wire now will be
m=πd2lρ4=π(3d)23lρ4=27πd2lρ4=27m (m=πd2lρ4)
Now the frequency becomes,
n=12×3l3T27m=19×12lTm=n9

So, the correct answer is “Option B”.

Additional Information:
Frequency is defined as the number of oscillations or occurrence per unit time. The unit of frequency is Hertz. Frequency is also defined as the reciprocal of time period. From the above calculation you can see that by increasing the length and diameter of the cross-section of wire the frequency decreases.

Note:
For problems like this, first calculate frequency in the original case, and then calculate the frequency for the changed case. Compare both of them to get a relationship between them. You can also calculate the time period because the time period is reciprocal of frequency.

Time period=1frequency

When the tension on the wire is large the wire will break. This breaking point is called breaking stress. So breaking stress is defined as the force on the wire per unit cross-sectional area.