(D) The kinetic energy per unit mass $(KE/m)$ has the dimensions of velocity squared,which is $[L^2 T^{-2}]$.We analyze the dimensions of the given op

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(D) The kinetic energy per unit mass $(KE/m)$ has the dimensions of velocity squared,which is $[L^2 T^{-2}]$.We analyze the dimensions of the given options:For option $D$,the expression is $\frac{\alpha \sigma}{\rho}$.The dimensions of tensile strength $\sigma$ (pressure) are $[M L^{-1} T^{-2}]$.The dimensions of density $\rho$ are $[M L^{-3}]$.Therefore,the dimensions of $\frac{\sigma}{\rho}$ are $\frac{[M L^{-1} T^{-2}]}{[M L^{-3}]} = [L^2 T^{-2}]$.Since $\alpha$ is a dimensionless constant,the expression $\frac{\alpha \sigma}{\rho}$ correctly represents the dimensions of kinetic energy per unit mass.

Class 11 Physics Units, Dimensions and Measurement Dimensional Analysis, Uses and Limitations

Difficult

$A$ flywheel can rotate in order to store kinetic energy. The flywheel is a uniform disk made of a material with a density $\rho$ and tensile strength $\sigma$ (measured in Pascals),a radius $r$,and a thickness $h$. The flywheel is rotating at the maximum possible angular velocity so that it does not break. Which of the following expressions correctly gives the maximum kinetic energy per kilogram that can be stored in the flywheel? Assume that $\alpha$ is a dimensionless constant.

A
$\alpha \sqrt{\frac{\rho \sigma}{r}}$
B
$\alpha h \sqrt{\frac{\rho \sigma}{r}}$
C
$\alpha \left( \frac{h}{r^2} \right) \left( \frac{\sigma}{\rho} \right)$
D
$\frac{\alpha \sigma}{\rho}$

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Units, Dimensions and Measurement

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Dimensional Analysis, Uses and Limitations

The frequency $(v)$ of an oscillating liquid drop may depend upon the radius $(r)$ of the drop,the density $(\rho)$ of the liquid,and the surface tension $(s)$ of the liquid as: $v = r^{a} \rho^{b} s^{c}$. The values of $a, b,$ and $c$ respectively are:

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If $E$,$M$,$L$,and $G$ denote energy,mass,angular momentum,and the universal gravitational constant respectively,then the quantity $\left(\frac{EL^{2}}{G^{2} M^{5}}\right)$ has the dimensions of:

MediumMHT CET 2020

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When a wave traverses a medium, the displacement of a particle located at $x$ at a time $t$ is given by $y = a \sin (bt - cx)$, where $a, b,$ and $c$ are constants of the wave. Which of the following is a quantity with dimensions?

MediumTS EAMCET 2009

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If mass is written as $m=kc^{p} G^{-1 / 2} \,h^{1 / 2}$ then the value of $P$ will be : (Constants have their usual meaning with $k$ a dimensionless constant)

DifficultJEE MAIN 2024

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The time dependence of a physical quantity $P$ is given by $P = P_0 \exp(-\alpha t^2)$,where $\alpha$ is a constant and $t$ is time. The constant $\alpha$:

EasyAIPMT 1993

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The frequency $(v)$ of an oscillating liquid drop may depend upon the radius $(r)$ of the drop,the density $(\rho)$ of the liquid,and the surface tension $(s)$ of the liquid as: $v = r^{a} \rho^{b} s^{c}$. The values of $a, b,$ and $c$ respectively are:

If $E$,$M$,$L$,and $G$ denote energy,mass,angular momentum,and the universal gravitational constant respectively,then the quantity $\left(\frac{EL^{2}}{G^{2} M^{5}}\right)$ has the dimensions of:

When a wave traverses a medium, the displacement of a particle located at $x$ at a time $t$ is given by $y = a \sin (bt - cx)$, where $a, b,$ and $c$ are constants of the wave. Which of the following is a quantity with dimensions?

If mass is written as $m=kc^{p} G^{-1 / 2} \,h^{1 / 2}$ then the value of $P$ will be : (Constants have their usual meaning with $k$ a dimensionless constant)

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