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 is not a valid application of 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?

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

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 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 incorrect according to the principle of homogeneity?

A hypothetical equation is given as $\rho = a \sqrt{P} + bT^2$, where $\rho$ is density, $P$ is pressure, and $T$ is temperature. The dimensions of $\frac{a}{b}$ 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?

The principle of homogeneity states that: