Dimensions of Physical Quantities and Dimensional Analysis

In the dimensional equation for the period of oscillation of a simple pendulum, $T = k g^a l^b$, where $T$ is the period, $g$ is the acceleration due to gravity, $l$ is the length of the pendulum, and $k$ is a dimensionless constant, what is the numerical value of the exponent $a$?

The gravitational force $F$ between two point masses $m_1$ and $m_2$ separated by a distance $r$ is expressed by the equation $F = \frac{G m_1 m_2}{r^2}$, where $G$ represents the universal gravitational constant. What is the dimensional formula for $G$?

The viscous force $F$ acting on a small sphere of radius $r$ moving with velocity $v$ through a liquid is given by Stokes' law, $F = 6\pi \eta r v$, where $\eta$ is the coefficient of viscosity. Which of the following expressions represents the base dimensions of $\eta$?

Power is defined as the rate at which work is done or energy is transferred. Which of the following expressions represents the correct dimensional formula for power?

Which of the following expressions represents the correct dimensions of pressure?

In atomic physics, the energy $E$ of a photon is related to its frequency $f$ by the formula $E = h f$, where $h$ is Planck's constant. What are the fundamental dimensions of Planck's constant $h$?

The speed of sound $v$ in a gas depends on the gas pressure $P$ and density $\rho$ according to the relation $v = k P^a \rho^b$, where $k$ is a dimensionless constant. What is the numerical value of the exponent $a$?

In electrostatics, Coulomb's law states that the force $F$ between two point charges $q_1$ and $q_2$ separated by a distance $r$ in a vacuum is given by $F = \frac{q_1 q_2}{4\pi \varepsilon_0 r^2}$, where $\varepsilon_0$ is the permittivity of free space. What are the fundamental dimensions of $\varepsilon_0$ expressed in terms of mass ($M$), length ($L$), time ($T$), and electric current ($I$)?

The volume flow rate $Q$ of a viscous liquid through a pipe depends on the radius $r$ of the pipe, the coefficient of viscosity $\eta$, and the pressure gradient $\frac{\Delta P}{L}$ according to the dimensional equation $Q = k r^x \eta^y \left(\frac{\Delta P}{L}\right)^z$, where $k$ is a dimensionless constant. What is the value of the exponent $x$?

The rate of heat transfer through a uniform metallic rod of cross-sectional area $A$ and length $d$ is given by $\frac{Q}{t} = \frac{k A \Delta T}{d}$, where $Q$ is heat energy, $t$ is time, $\Delta T$ is temperature difference, and $k$ is thermal conductivity. What is the dimensional formula for $k$ in terms of mass ($M$), length ($L$), time ($T$), and temperature ($\Theta$)?

The fundamental frequency $f$ of a stretched vibrating string depends on the tension force $F$, the length of the string $l$, and its linear mass density $\mu$ (mass per unit length) according to the relation $f = k F^a l^b \mu^c$, where $k$ is a dimensionless constant. Which of the following sets of exponents $(a, b, c)$ correctly satisfies dimensional homogeneity?

The dynamic pressure $P$ exerted by a moving fluid depends on its density $\rho$ and flow velocity $v$ according to the relationship $P = k \rho^a v^b$, where $k$ is a dimensionless constant. What is the value of the exponent $b$?

Newton's law of universal gravitation states that the gravitational force $F$ between two point masses $m_1$ and $m_2$ separated by a distance $r$ is given by $F = \frac{G m_1 m_2}{r^2}$, where $G$ is the universal gravitational constant. What is the dimensional formula of $G$?