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QJEEM23Phab1d: JEE Main 2023 Physics - Class 11 JEEM23Phab1d Solution: The speed of a wave produced in water is given by . Where and…

The speed of a wave produced in water is given by v=λagbρcv=\lambda^{a} g^{b} \rho^{c} . Where λ,g\lambda, g and ρ\rho are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a,ba, b and cc respectively, are :

Multiple Choice Options

  • A 12,0,12\frac{1}{2}, 0, \frac{1}{2}
  • B 1,1,01,1,0
  • C 1,−1,01,-1,0
  • D 12,12,0\frac{1}{2}, \frac{1}{2}, 0

Step-by-Step Verified Solution

amp;v=λagbρcamp;using dimension formulaamp;⇒[M0 L1 T−1]=[L1]a[L1 T−2]b[M1 L−3]camp;⇒[M0 L1 T−1]=[McLa+b−cT−2 b]amp;∴c=0,a+b−3c=1,−2 b=−1⇒b=12amp; Now a+b−3c=1amp;⇒a+12−0=1amp;⇒a=12amp;∴a=12, b=12,c=0\begin{aligned}& v=\lambda^{\mathrm{a}} \mathrm{g}^{\mathrm{b}} \rho^{\mathrm{c}} \\\\& \text {using dimension formula} \\\\& \Rightarrow\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^{-1}\right]=\left[\mathrm{L}^1\right]^{\mathrm{a}}\left[\mathrm{L}^1 \mathrm{~T}^{-2}\right]^{\mathrm{b}}\left[\mathrm{M}^1 \mathrm{~L}^{-3}\right]^{\mathrm{c}} \\\\& \Rightarrow\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^{-1}\right]=\left[\mathrm{M}^{\mathrm{c}} \mathrm{L}^{\mathrm{a}+\mathrm{b}-\mathrm{c}} \mathrm{T}^{-2 \mathrm{~b}}\right] \\\\& \therefore \mathrm{c}=0, \mathrm{a}+\mathrm{b}-3 \mathrm{c}=1,-2 \mathrm{~b}=-1 \Rightarrow \mathrm{b}=\frac{1}{2} \\\\& \text { Now } \mathrm{a}+\mathrm{b}-3 \mathrm{c}=1 \\\\& \Rightarrow \mathrm{a}+\frac{1}{2}-0=1 \\\\& \Rightarrow \mathrm{a}=\frac{1}{2} \\\\& \therefore \mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{1}{2}, \mathrm{c}=0\end{aligned}

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