$A$ ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$,given by $t = A \rho^{a} r^{b} \eta^{c} \sigma^{d}$,where $A$ is a constant and $a, b, c, d$ are integers. The value of $\frac{b+c}{a+d}$ is . . . . . . .

Applying the principle of homogeneity of dimensions,determine which one is correct. Where $T$ is time period,$G$ is gravitational constant,$M$ is mass,and $r$ is the radius of the orbit.

$A, B, C$ and $D$ are four different physical quantities having different dimensions. None of them is dimensionless. We know that the equation $AD = C \ln(BD)$ holds true. Which of the following combinations is not a meaningful quantity?

The refractive index of the medium is $\mu = A + \frac{B}{\lambda^{2}}$,where $A$ and $B$ are constants and $\lambda$ is the wavelength of light. The dimensions of $B$ are the same as that of

In terms of potential difference $V$,electric current $I$,permittivity $\varepsilon_0$,permeability $\mu_0$,and speed of light $c$,the dimensionally correct equation$(s)$ is(are):
$(A)$ $\mu_0 I^2 = \varepsilon_0 V^2$
$(B)$ $\varepsilon_0 I = \mu_0 V$
$(C)$ $I = \varepsilon_0 cV$
$(D)$ $\mu_0 cI = V$

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.