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 $a$?

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?

The period $T$ of oscillation of a spring-mass system depends on the mass $m$ and the spring constant $k$. Using dimensional analysis and the principle of homogeneity, determine the relationship between $T$, $m$, and $k$.

A highly advanced alien civilization uses a unit of mass called a 'glorp' and a unit of length called a 'blarp'. Their fundamental unit of time is the 'zorp'. They discover a fundamental law of physics relating force (F), mass (m), length (l), and time (t) expressed as $F = K m^a l^b t^c$, where K is a dimensionless constant. If the dimensions of force in their system are glorp\cdot blarp \cdot zorp^{-2}$, what are the values of a, b, and c?

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 $b$?

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 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 inconsistent?

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

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 $a$, $b$.