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?

$A$ 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.

$A$ new system of units is proposed in which the unit of mass is $\alpha \ kg$,the unit of length is $\beta \ m$,and the unit of time is $\gamma \ s$. How much will $5 \ J$ measure in this new system?

If the velocity of light $C$, the gravitational constant $G$, and Planck's constant $h$ are chosen as the fundamental units, the dimension of density in the new system is

Stokes' law states that the viscous drag force $F$ experienced by a sphere of radius $a$,moving with a speed $v$ through a fluid with coefficient of viscosity $\eta$,is given by $F=6 \pi \eta a v$. If this fluid is flowing through a cylindrical pipe of radius $r$,length $l$ and pressure difference of $p$ across its two ends,then the volume of water $V$ which flows through the pipe in time $t$ can be written as $\frac{V}{t}=k\left(\frac{p}{l}\right)^a \eta^b r^c$,where $k$ is a dimensionless constant. Correct values of $a, b$ and $c$ are

Velocity $(v)$ and acceleration $(a)$ in two systems of units $1$ and $2$ are related as $v_{2} = \frac{n}{m^{2}} v_{1}$ and $a_{2} = \frac{a_{1}}{mn}$ respectively. Here $m$ and $n$ are constants. The relations for distance $(L)$ and time $(T)$ in the two systems are: