The viscous force $F$ acting on a sphere of radius $r$ moving with velocity $v$ through a fluid of viscosity $\eta$ is given by $F = 6\pi \eta rv + kr^2v^2$. The dimensions of $k$ are:

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

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

The viscous force $F$ acting on a sphere of radius $r$ moving with velocity $v$ through a fluid of viscosity $\eta$ is given by $F = 6\pi \eta rv + kr^2v^2$. The dimensions of $k$ are:

The frequency $ν$ of vibration of a stretched string depends on its length $l$, its linear mass density $μ$, and its tension $T$. Which of the following is a possible formula for $ν$, consistent with the principle of homogeneity?

In the equation $F = ma$, if the unit of force ($F$) is newton (N), mass ($m$) is kilogram (kg), and acceleration ($a$) is $ms^{-2}$, is the equation dimensionally homogeneous?

Which of the following equations is dimensionally incorrect according to 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 $abc$?

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?

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?

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