The time period ($T$) of a simple pendulum depends on its length ($l$) and acceleration due to gravity ($g$). Which of the following is dimensionally consistent with the principle of homogeneity?

Given the equation $P = \frac{a}{V} - \frac{b}{V^2} + cT$, where $P$ is pressure, $V$ is volume, and $T$ is temperature. What are the dimensions of $abc$?

The velocity of a wave is given by $v = \sqrt{\frac{T}{\mu}} + ax^2$, where $T$ is tension, $\mu$ is linear density, and $x$ is displacement. What are the dimensions of $a$?

The viscous force $F$ acting on a sphere moving through a liquid depends on the radius $r$ of the sphere, the velocity $v$ of the sphere, and the coefficient of viscosity $η$ of the liquid. If $F = kη^arv^b$, where $k$ is a dimensionless constant, using dimensional analysis, find the values of $a$, $b$.

Which of the following equations is dimensionally inconsistent?

If the equation $P = \frac{F}{A}$ is dimensionally homogeneous, where $P$ is pressure, $F$ is force, and $A$ is area, what are the dimensions of pressure?

Which of the following is not a valid application of the principle of homogeneity?

If $x = a + bt^2$, where $x$ is in meters and $t$ is in seconds, what are the dimensions of $b$?

If the velocity ($v$), acceleration ($a$), and time ($t$) are related by an equation $v = at + bt^2$, what are the dimensions of $b$?

Which of the following equations is dimensionally homogeneous?

Which of the following equations is dimensionally incorrect according to the principle of homogeneity?