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

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

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 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:

The velocity of water waves \(𝑣\) may depend upon their wavelength \(𝜆\), the density of water \(𝜌\) 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 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 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) $1,2,\frac{1}{3}$

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

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: