The density of a material in $CGS$ system of units  is $4\, g\, cm^{-3}$ . In a system of units in which unit of length is $10\, cm$ and unit of mass is $100 \,g,$ the value of density of material will be

If time $(t)$, velocity $(u)$, and angular momentum $(I)$ are taken as the fundamental units. Then the dimension of mass $({m})$ in terms of ${t}, {u}$ and ${I}$ is

The equation of state of a real gas is given by $\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}$, where $\mathrm{P}, \mathrm{V}$ and $\mathrm{T}$ are pressure. volume and temperature respectively and $R$ is the universal gas constant. The dimensions of $\frac{a}{b^2}$ is similar to that of :

Turpentine oil is flowing through a tube of length $l$ and radius $r$. The pressure difference between the two ends of the tube is $P .$ The viscosity of oil is given by $\eta=\frac{P\left(r^{2}-x^{2}\right)}{4 v l}$ where $v$ is the velocity of oil at a distance $x$ from the axis of the tube. The dimensions of $\eta$ are

A book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:

$(a)\;y=a \sin \left(\frac{2 \pi t}{T}\right)$

$(b)\;y=a \sin v t$

$(c)\;y=\left(\frac{a}{T}\right) \sin \frac{t}{a}$

$(d)\;y=(a \sqrt{2})\left(\sin \frac{2 \pi t}{T}+\cos \frac{2 \pi t}{T}\right)$

$(a=$ maximum displacement of the particle, $v=$ speed of the particle. $T=$ time-period of motion). Rule out the wrong formulas on dimensional grounds.

If force $[F],$ acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical quantities. Find the dimensions of energy.