The velocity of a freely falling body changes as $g^p{h}^q$ where g is acceleration due to gravity and h is the height. The values of p and q are 

(1) $1,\frac{1}{2}$

(2) $\frac{1}{2},\frac{{\displaystyle 1}}{2}$

(3) $\frac{1}{2},1$

(4) 1, 1

The equation of a wave is given by $Y=A\sin \omega (\frac{x}{v}-k)$ where $\omega$ is the angular velocity, x is length and $v$ is the linear velocity. The dimension of k is 

(1) LT

(2) T

(3) $T^{-1}$

(4) T²

The velocity of water waves $v$ may depend upon their wavelength $\lambda$, the density of water $\rho$ and the acceleration due to gravity g. The method of dimensions gives the relation between these quantities as 

(1) $v^2\propto g{\lambda }^{-1}{\rho }^{-1}$

(2) $v^2\propto g\lambda \rho$

(3) $v^2\propto g\lambda$

(4) $v^2\propto g^{-1}{\lambda }^{-3}$

The frequency of vibration \(f\) of a mass \(m\) suspended from a spring of spring constant \(k\) is given by a relation of type \(f= Cm^{x}k^{y}\); where \(C\) is a dimensionless quantity. The values of \(x\) and \(y\) will be:
1. \(x=\frac{1}{2},~y= \frac{1}{2}\)
2. \(x=-\frac{1}{2},~y= -\frac{1}{2}\)
3. \(x=\frac{1}{2},~y= -\frac{1}{2}\)
4. \(x=-\frac{1}{2},~y= \frac{1}{2}\)

The period of a body under SHM i.e. presented by $T=P^a{D}^b{S}^c$; where P is pressure, D is density and S is surface tension. The value of a, b and c are 

(1) $-\frac{3}{2},\frac{1}{2},1$

(2) $-1,-2,3$

(3) $\frac{1}{2},-\frac{3}{2},-\frac{1}{2}$

(4) $1,2,\frac{1}{3}$

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 \((\text{i})\) the mass of the ball \(m\) \((\text{ii})\) \(\eta\) \((\text{iii})\) \(r\) and \((\text{iv})\) acceleration due to gravity \(g\). Which of the following relations is dimensionally correct:

$The<mspace\; width="1em"></mspace>measurement<mspace\; width="1em"></mspace>are<mspace\; width="1em"></mspace>made<mspace\; width="1em"></mspace>as<mspace\; width="1em"></mspace>18.425<mspace\; width="1em"></mspace>cm,<mspace\; width="1em"></mspace>7.21<mspace\; width="1em"></mspace>cm<mspace\; width="1em"></mspace>and<mspace\; width="1em"></mspace>5.0<mspace\; width="1em"></mspace>cm.<mspace\; width="1em"></mspace>The<mspace\; width="1em"></mspace>addition$
$should<mspace\; width="1em"></mspace>be<mspace\; width="1em"></mspace>written<mspace\; width="1em"></mspace>as:$
$\left(a\right)<mspace\; width="1em"></mspace>30.635<mspace\; width="1em"></mspace>cm$
$\left(b\right)<mspace\; width="1em"></mspace>30.64<mspace\; width="1em"></mspace>cm$
$\left(c\right)<mspace\; width="1em"></mspace>30.63<mspace\; width="1em"></mspace>cm$
$\left(d\right)<mspace\; width="1em"></mspace>30.6<mspace\; width="1em"></mspace>cm$

The quantities \(A\) and \(B\) are related by the relation, \(m= \frac{A}{B}\), where \(m\) is the linear density and \(A\) is the force. The dimensions of \(B\) are of:

A spherical body of mass m and radius r is allowed to fall in a medium of viscosity $\eta$. The time in which the velocity of the body increases from zero to 0.63 times the terminal velocity $\left(v\right)$ is called time constant $\left(\tau \right)$. Dimensionally $\tau$ can be represented by 

(1) $\frac{m{r}^2}{6\pi \eta }$

(b) $\sqrt{\left(\frac{6\pi mr\eta }{g^2}\right)}$

(c) $\frac{m}{6\pi \eta rv}$

(4) None of the above