In a particular system,the units of length,mass,and time are chosen to be $10 \; cm$,$10 \; g$,and $0.1 \; s$ respectively. The unit of force in this system will be equal to (in $; N$)?

$A$ famous relation in physics relates 'moving mass' $m$ to the 'rest mass' $m_{0}$ of a particle in terms of its speed $v$ and the speed of light $c$. (This relation first arose as a consequence of special relativity due to Albert Einstein). $A$ boy recalls the relation almost correctly but forgets where to put the constant $c$. He writes:
$m = \frac{m_{0}}{(1 - v^{2})^{1/2}}$
Guess where to put the missing $c$.

Consider a simple pendulum,having a bob attached to a string,that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length $(l)$,mass of the bob $(m)$,and acceleration due to gravity $(g)$. Derive the expression for its time period using the method of dimensions.

The position of a particle at time $t$ is given by the relation $x(t) = \left( \frac{v_0}{\alpha} \right) (1 - e^{-\alpha t})$,where $v_0$ is a constant and $\alpha > 0$. The dimensions of $v_0$ and $\alpha$ are respectively:

$A$ massive black hole of mass $m$ and radius $R$ is spinning with angular velocity $\omega$. The power $P$ radiated by it as gravitational waves is given by $P=G c^{-5} m^{x} R^{y} \omega^{z}$,where $c$ and $G$ are the speed of light in free space and the universal gravitational constant,respectively. Then,

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 the coefficient of viscosity of the liquid having the dimensional formula $[M^1 L^{-1} T^{-1}]$. Check whether the equation is dimensionally correct.