Question

The damping force of an oscillating particle is observed to be proportional to velocity. The constant of proportionality can be measured in
A) \[{\text{kg}}{{\text{s}}^{{\text{ - 1}}}}\]
B) \[{\text{kgs}}\]
C) ${\text{kgm}}{{\text{s}}^{{\text{ - 1}}}}$
D) \[{\text{kg}}{{\text{m}}^{{\text{ - 1}}}}{{\text{s}}^{{\text{ - 1}}}}\]

Answer 1

Hint: We know that the force is proportional to the velocity of the particle. We will use the concepts of dimensional formula to determine the units of constant of proportionality.

Complete step by step answer:
We’ve been given that the damping force of an oscillating particle is observed to be proportional to velocity. We know that the dimension of force are \[kgm{s^{ - 2}}\]. This can be written in the dimensional formula as
$[F] = {M^1}{L^1}{T^{ - 2}}$
And as the force is proportional to the velocity of the particle,
$F \propto v$
The proportionality of constant $(b)$ can be written as
$F = bv$
Then the dimensional formula of velocity are $[v] = {L^1}{T^{ - 1}}$ and its units are $m{s^{ - 1}}$. Then the dimensional formula of $b$ will be
$[b] = \dfrac{{[F]}}{{[v]}}$
Which can be written as
$[b] = \dfrac{{{M^1}{L^1}{T^{ - 2}}}}{{{L^1}{T^{ - 1}}}}$
Hence the dimensional formula of the proportionality constant will be
$[b] = {M^1}{T^{ - 1}}$
The units of the proportionality constant will hence be $kg{s^{ - 1}}$ which corresponds to option (A).
Additional information:
Depending on the value of the proportionality constant, the motion of the oscillating particle will be different. The motion can be over-damped which means that the damping constant is strong enough to not let the particle oscillate. It can also be under-damped which means that the particle will oscillate with decreasing amplitude. This is because the energy of the particle will be lost in the damper. The rate of loss of energy will depend on the value of the proportionality or the damping constant.

Note: The damping force on an oscillating particle is often treated like a drag force that slows down the oscillating particle. The friction force depends on the strength of the damping constant whose dimensional formula we found in the above equation.