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Question
Choose the correct option.
The tension in a piano wire is increased by 25%. Its frequency becomes ______ times the original frequency.
Options
0.8
1.12
1.25
1.56
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Solution
The tension in a piano wire is increased by 25%. Its frequency becomes 1.12 times the original frequency.
Explanation:
Express the equation for frequency.
`f = v/lambda`
Here, f is frequency, v is velocity and λ is the wavelength.
Express the equation for old velocity for the stretched string.
v = `sqrt(T/m)`
Here, T is old Tension and m is the mass density.
Express the equation for the new velocity for the stretched string.
`v_0 = sqrt(T_0/m)` ......(1)
Here, T0 is old Tension and m is the mass density.
Tension is increased by 25%. So express the equation for new tension.
T0 = 1.25T,
Here T0 is new tension and T is old tension.
Substitute 1.25 T for T0 in the equation in (1).
`v_0 = sqrt((1.25T)/m)`
= `sqrt(1.25) xx sqrt(T/m)`
= `1.118 xx sqrt(T/m)`
Substitute v for `sqrt(T/m)` to obtain new velocity.
`v_0 = 1.118 xx v`
= 1.12 v
Therefore, the frequency becomes 1.12 times the original frequency.
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