The volume of a liquid flowing out per second of a pipe of length $l$ and radius $r$ is written by a student as $V\, = \,\frac{{\pi p{r^4}}}{{8\eta l}}$ where $p$ is the pressure difference between the two ends of the pipe and $\eta $ is coefficent of viscosity of the liquid having dimensional formula $[M^1L^{-1}T^{-1}] $. Check whether the equation is dimensionally correct.
The volume of a liquid flowing out per second of a pipe of length $l$ and radius $r$ is written by a student as $V\, = \,\frac{{\pi p{r^4}}}{{8\eta l}}$ where $p$ is the pressure difference between the two ends of the pipe and $\eta $ is coefficent of viscosity of the liquid having dimensional formula $[M^1L^{-1}T^{-1}] $. Check whether the equation is dimensionally correct.
The volume of a liquid flowing out per second of a pipe is given by $\mathrm{V}=\frac{\pi}{8} \frac{p r^{4}}{\eta l}$
${[\mathrm{V}]=\frac{[\text { Volume }]}{[\mathrm{Time}]}=\frac{\left[\mathrm{L}^{3}\right]}{[\mathrm{T}]}=\left[\mathrm{L}^{3} \mathrm{~T}^{-1}\right]}$
${[p]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]}$
${[\eta]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]}$
${[l]=[\mathrm{L}]}$
${[r]=[\mathrm{L}]}$
${[\mathrm{LHS}]=[\mathrm{V}]=\frac{\left[\mathrm{L}^{3}\right]}{[\mathrm{T}]}=\left[\mathrm{L}^{3} \mathrm{~T}^{-1}\right]}$
${[\mathrm{RHS}]=\frac{\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right] \times\left[\mathrm{L}^{4}\right]}{\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right] \times[\mathrm{L}]}=\left[\mathrm{L}^{3} \mathrm{~T}^{-1}\right]}$
${[\mathrm{LHS}]=[\mathrm{RHS}]}$
Thus, equation is dimensionally correct.
Similar Questions
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A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material. Boltzmann constant $\left(k_B\right)$, the absolute temperature $(T)$, the number per unit volune $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particless. Which of the following expression($s$) for $l$ is(are) dimensionally correct?
($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$
A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material. Boltzmann constant $\left(k_B\right)$, the absolute temperature $(T)$, the number per unit volune $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particless. Which of the following expression($s$) for $l$ is(are) dimensionally correct?
($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$
- [IIT 2016]
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