What is the constant of propotionality of an oscillator if the damping force is directly propotional to the velocity

The correct option is kg.s^-1.

Let us derive this step-by-step using dimensional analysis and the definition of the damping force.

The problem states that the damping force ($F$) acting on an oscillator is directly proportional to its velocity ($v$). Mathematically, we can write this relationship as:

$F=-bv$

where $b$ is the constant of proportionality (known as the damping coefficient). The negative sign indicates that the damping force opposes the direction of motion, which does not affect the units.

To find the units of the constant of proportionality $b$, we rearrange the formula to solve for $b$:

$b=\frac{F}{v}$

Now, let us substitute the SI base units of force ($F$) and velocity ($v$):
1. The SI unit of force is the Newton (N), which is defined in SI base units as kilogram meters per second squared:
$\left[F\right]=\text{kg}·\text{m}·{\text{s}}^{-2}$
2. The SI unit of velocity is meters per second:
$\left[v\right]=\text{m}·{\text{s}}^{-1}$

Substituting these units back into the equation for $b$:

$\left[b\right]=\frac{\text{kg}·\text{m}·{\text{s}}^{-2}}{\text{m}·{\text{s}}^{-1}}$

Simplifying the fraction by canceling out the unit of length (meters, m) from both the numerator and the denominator, we get:

$\left[b\right]=\text{kg}·{\text{s}}^{-2-(-1)}=\text{kg}·{\text{s}}^{-1}$

Thus, the SI unit for the constant of proportionality of the damping force is kg.s^-1.