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**Hint:**Try to convert the above question in mathematical form and then find the unit by putting their equivalent units in the mathematical expression.

**Complete step by step answer:**

Let's say the damping force of the oscillator is \[F\] and the applied velocity on it is \[V\].

According to the question the damping force on the oscillator is directly proportional to the velocity.

Then we can say that:

\[F \propto V\].

If we try to convert this mathematical expression into the equation, then we need to multiply it by a constant.

Let's say, the constant of the proportionality is \[K\].

So, the above equation will become \[F = (K \times V)\] .

Damping force is nothing but force. Force is the productivity of mass and acceleration is applicable on the object. Acceleration is change in velocity with unit time.

To implement the unit of the acceleration we need the unit of velocity as well as the unit of time.

So, we get \[ \text{Acceleration} = \dfrac{ \text{velocity}}{ \text{time}}\] and \[\text{Velocity }= \dfrac{ \text{Displacement}}{ \text{time}}\] .

So, we can write it as,

\[\text{Unit of Velocity} = \dfrac{\text{unit of displacement}}{\text{unit of time}}\]

As the above options we will try to find the unit in S.I form.

So, the unit of displacement in S.I is meter \[(m)\] and the unit of time in S.I is second \[(s)\] and unit of mass in S.I is kilogram\[(kg)\].

So, \[\text{Unit of Velocity} = \dfrac{\text{meter}}{\text{second}} = \dfrac{m}{s}\] .

Again, \[\text{Unit of Acceleration} = \dfrac{\text{unit of Velocity}}{\text{unit of time}} = \dfrac{{\dfrac{\text{meter}}{\text{second}}}}{\text{second}} = \dfrac{\text{meter}}{\text{second^2}} = \dfrac{m}{{{s^2}}}\] .

Again, \[\text{Unit of Force} = (\text{Unit of Mass} \times \text{Unit of Acceleration})\].

\[\text{Unit of Force} = (\text{Kilogram} \times \dfrac{\text{meter}}{\text{second^2}}) = (kgms^{ - 2})\] .

So, the unit of proportionality is the unit of force divided by the unit of velocity.

So, \[\text{Unit of proportionality} = \dfrac{\text{unit of force}}{\text{unit of velocity}}\] .

Or, \[\text{Unit of proportionality} = \dfrac{{kgm{s^{ - 2}}}}{{m{s^{ - 1}}}} = \dfrac{{kgm}}{{{s^2}}} \times \dfrac{s}{m} = \dfrac{{kg}}{s} = kg{s^{ - 1}}\] .

**Hence, the correct answer is option (A).**

**Note:**A unit of measurement is the definite magnitude of a quantity. So, each quantity has a different measurement process as well as a different measurement unit.

There are two ways to measure units:

• S.I (System International)

• C.G.S (Centimeter-Gram-Second)

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