A physical parameter '\(a\)' can be determined by measuring the parameters \(b\), \(c\), \(d\) and \(e\) using the relation, \(a= \dfrac{b^{\alpha}c^{\beta}}{d^{\gamma}e^{\delta}}.\) If the maximum errors in the measurement of \(b, ~c, ~d,~\text{and}~e\) are \(b_1\%,~c_1\%,~d_1\%~\text{and}~e_1\%\), then the maximum error in the value of '\(a\)' determined by the experiment is:
1. \((b_1+c_1+d_1+e_1)\%\)
2. \((b_1+c_1-d_1-e_1)\%\)
3. \((\alpha b_1+\beta c_1-\gamma d_1-\delta e_1)\%\)
4. \((\alpha b_1+\beta c_1+\gamma d_1+\delta e_1)\%\)

The unit of angular acceleration in the SI system is 

(1) $Nk{g}^{-1}$

(2) $m{s}^{-2}$

(3) $rad{s}^{-2}$

(4) $mk{g}^{-1}K$

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 this type $f=C{m}^x{K}^y$; where C is a dimensionless quantity. The value of x and y are 

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 = A/B, where m is the linear density and A is the force. The dimensions of B are of

1. Pressure

2. Work

3. Latent heat

4. None of the above

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. ${\mathrm{v}}^2\propto {\mathrm{g\lambda }}^{-1}{\mathrm{\rho }}^{-1}$

2. ${\mathrm{v}}^2\propto \mathrm{g\lambda \rho }$

3. ${\mathrm{v}}^2\propto \mathrm{g\lambda }$

4. ${\mathrm{v}}^2\propto {\mathrm{g}}^{-1}{\mathrm{\lambda }}^{-3}$

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 is presented by $\mathrm{T}={\mathrm{P}}^{\mathrm{a}}{\mathrm{D}}^{\mathrm{b}}{\mathrm{S}}^{\mathrm{c}}$; 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 according to $g^p{h}^q$ 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 (i) the mass of the ball m, (ii) $\eta$, (iii) r and (iv) acceleration due to gravity g. Which of the following relations is dimensionally correct?

1. $v_T\propto \frac{mg}{\eta r}$

2. $v_T\propto \frac{\eta r}{mg}$

3. $v_T\propto \eta rmg$

4. $v_T\propto \frac{mgr}{\eta }$