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

Last updated date: 08th Sep 2024
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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.

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).