A force defined by $F=\alpha t^2+\beta t$ acts on a particle at a given time $t$. The factor which is dimensionless, if $\alpha$ and $\beta$ are constants, is:

  • [NEET 2024]
  • A

    $\alpha t / \beta$

  • B

    $\alpha \beta t$

  • C

    $\alpha \beta / t$

  • D

    $\beta t / \alpha$

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In Vander Waals equation $\left[ P +\frac{ a }{ V ^{2}}\right][ V - b ]= RT$; $P$ is pressure, $V$ is volume, $R$ is universal gas constant and $T$ is temperature. The ratio of constants $\frac{a}{b}$ is dimensionally equal to .................

  • [JEE MAIN 2022]

Given that $\int {{e^{ax}}\left. {dx} \right|}  = {a^m}{e^{ax}} + C$, then which statement is incorrect (Dimension of $x =  L^1$) ?

Consider two physical quantities A and B related to each other as $E=\frac{B-x^2}{A t}$ where $E, x$ and $t$ have dimensions of energy, length and time respectively. The dimension of $A B$ is

  • [JEE MAIN 2024]

A small steel ball of radius $r$ is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity $\eta $. After some time the velocity of the ball attains a constant value known as terminal velocity ${v_T}$. The terminal velocity depends on $(i)$ the mass of the ball $m$, $(ii)$ $\eta $, $(iii)$ $r$ and $(iv)$ acceleration due to gravity $g$. Which of the following relations is dimensionally correct

Match List $I$ with List $II$ and select the correct answer using the codes given below the lists :

List $I$ List $II$
$P.$ Boltzmann constant $1.$ $\left[ ML ^2 T ^{-1}\right]$
$Q.$ Coefficient of viscosity $2.$ $\left[ ML ^{-1} T ^{-1}\right]$
$R.$ Planck constant $3.$ $\left[ MLT ^{-3} K ^{-1}\right]$
$S.$ Thermal conductivity $4.$ $\left[ ML ^2 T ^{-2} K ^{-1}\right]$

Codes: $ \quad \quad P \quad Q \quad R \quad S $ 

  • [IIT 2013]