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Question: The frequency \((n)\) of a tuning fork depends upon the length\(\left( L \right)\) of its prongs, th...

The frequency (n)(n) of a tuning fork depends upon the length(L)\left( L \right) of its prongs, the density (d)\left( d \right) and Young’s modulus (Y)\left( Y \right) of its material. It is given that n∝LadbYcn \propto {L^a}{d^b}{Y^c}. The values of aa, bb and cc are:
(A) 1,12,−121,\dfrac{1}{2}, - \dfrac{1}{2}
(B) −1,−12,12 - 1, - \dfrac{1}{2},\dfrac{1}{2}
(C) 12,−1,−12\dfrac{1}{2}, - 1, - \dfrac{1}{2}
(D) 12,−12,1\dfrac{1}{2}, - \dfrac{1}{2},1

Explanation

Solution

Hint To solve this question, we need to perform the dimensional analysis for the given relation between the quantities in the question. To do this, we have to consider the dimensions of the quantities on the LHS and RHS and equate separately the powers of the dimensions to get the equations corresponding to the unknown variables.

Complete step by step answer
According to the question, the frequency (n)(n) is proportional to the length LL raised to the poweraa, density dd raised to the power bb and the Young’s modulus YY raised to the power cc, that is,
⇒n∝LadbYc\Rightarrow n \propto {L^a}{d^b}{Y^c}
Removing the proportionality sign with the constantcc, we get
⇒n=c(LadbYc)\Rightarrow n = c\left( {{L^a}{d^b}{Y^c}} \right) (1)
For the above equation to be correct, the dimensions of the quantity in the LHS should be equal to the dimensions of the quantities in the RHS.
Replacing the quantities of the above equation with their dimensions, we get
⇒[n]=[M0L0T−1]\Rightarrow \left[ n \right] = \left[ {{M^0}{L^0}{T^{ - 1}}} \right], [L]=[M0L1T0]\left[ L \right] = \left[ {{M^0}{L^1}{T^0}} \right], [d]=[M1L−3T0]\left[ d \right] = \left[ {{M^1}{L^{ - 3}}{T^0}} \right] and [Y]=[M1L−1T−2]\left[ Y \right] = \left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]
∵c\because cis a constant, so it has no dimensions.
Substituting these in (1) we get
⇒[M0L0T−1]=[M0L1T0]a[M1L−3T0]b[M1L−1T−2]c\Rightarrow \left[ {{M^0}{L^0}{T^{ - 1}}} \right] = {\left[ {{M^0}{L^1}{T^0}} \right]^a}{\left[ {{M^1}{L^{ - 3}}{T^0}} \right]^b}{\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]^c}
⇒[M0L0T−1]=[M0LaT0][MbL−3bT0][McL−cT−2c]\Rightarrow \left[ {{M^0}{L^0}{T^{ - 1}}} \right] = \left[ {{M^0}{L^a}{T^0}} \right]\left[ {{M^b}{L^{ - 3b}}{T^0}} \right]\left[ {{M^c}{L^{ - c}}{T^{ - 2c}}} \right]
On simplifying, we get
⇒[M0L0T−1]=[Mb+cLa−3b−cT−2c]\Rightarrow \left[ {{M^0}{L^0}{T^{ - 1}}} \right] = \left[ {{M^{b + c}}{L^{a - 3b - c}}{T^{ - 2c}}} \right]
Comparing the exponents of similar dimensions, we get
⇒b+c=0\Rightarrow b + c = 0 (2)
⇒a−3b−c=0\Rightarrow a - 3b - c = 0 (3)
And
⇒−2c=−1\Rightarrow - 2c = - 1 (4)
From (4), we get c=12c = \dfrac{1}{2}
Putting this in (2)
⇒b+12=0\Rightarrow b + \dfrac{1}{2} = 0
⇒b=−12\Rightarrow b = - \dfrac{1}{2}
Putting the values ofb,cb,cin (3)
⇒a+32−12=0\Rightarrow a + \dfrac{3}{2} - \dfrac{1}{2} = 0
⇒a+1=0\Rightarrow a + 1 = 0
Finally, a=−1a = - 1
∴a=−1,b=−12,c=12\therefore a = - 1,b = - \dfrac{1}{2},c = \dfrac{1}{2}
Putting these values in (1)
⇒n=c(L−1d−12Y12)\Rightarrow n = c\left( {{L^{ - 1}}{d^{ - \dfrac{1}{2}}}{Y^{\dfrac{1}{2}}}} \right)
Or, f=cLYdf = \dfrac{c}{L}\sqrt {\dfrac{Y}{d}}
Therefore, the formula for the frequency of a tuning fork is
⇒f=cLYd\Rightarrow f = \dfrac{c}{L}\sqrt {\dfrac{Y}{d}} , where ccis a constant.
So, the values of aa, bb and cc are −1,−12,12 - 1, - \dfrac{1}{2},\dfrac{1}{2} respectively.
Hence, the correct answer is option B.

Note
The dimensions of all the quantities of the question have been deduced using their respective fundamental formulae. We can use any formula in physics corresponding to a quantity to find out the dimension of that quantity.