Practice Practice Questions

Question 1

The frequency of vibration $f$ of a mass $m$ suspended from a spring of spring constant $k$ is given by a relation $f = C \sqrt{\frac{k}{m}}$, where $C$ is a dimensionless constant. The values of $x$…

Accepted Answer

From dimensional analysis, $f = C \sqrt{\frac{k}{m}}$. Solving for $x$ and $y$, we get $x = \frac{1}{2}, y = -\frac{1}{2}$.

Question 2

Which two of the following five physical parameters have the same dimensions? (i) Energy density (ii) Refractive index (iii) Dielectric constant (iv) Young's modulus (v) Magnetic field

Accepted Answer

Energy density and Young's modulus both have dimensions $[M L^{-1} T^{-2}]$. Refractive index and dielectric constant are dimensionless.

Question 3

If $p$ represents radiation pressure, $c$ represents speed of light, and $S$ represents radiation energy striking unit area per sec. The non-zero integers $x, y, z$ such that $pS^x c^y = z$ are

Accepted Answer

Radiation pressure $p = \frac{S}{c}$. Energy striking unit area per second $S = [ML^{-1}T^{-3}]$. Solving equations $x + y = 0$, $-x + z = 0$, and $-2x - 3y - z = 0$, we get $x = 1, y = -2, z = 1$.

Question 4

Dimensional formula of self-inductance is

Accepted Answer

As we know that emf induced in the inductor is given by $e = -L \frac{di}{dt}$, $L = \frac{e}{\frac{di}{dt}}$. Substituting dimensions, $L = \frac{[ML^{2}T^{-3}A^{-1}]}{[T^{-1}A]} = [ML^{2}T^{-2}A^{-2}]$.

Question 5

If force (F), velocity (v), and time (T) are taken as fundamental quantities, then the dimensions of mass are

Accepted Answer

We know that $F = ma$. Substituting $a = v/T$, we derive $[m] = [F v^{-1} T^{-1}]$.

Practice

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