The damping force on an oscillator is directly proportional to the velocity. The units of the constant to proportionality are

Options

A. `kg ms^(-1)`
B. `kg ms^(-2)`
C. `kg s^(-1)`
D. `kg s`

Answer

To find the units of the constant of proportionality for the damping force that is directly proportional to velocity, we can follow these steps:

### Step 1: Understand the relationship
The damping force \( F_d \) is directly proportional to the velocity \( v \). This can be expressed mathematically as:
\[ F_d = C \cdot v \]
where \( C \) is the constant of proportionality.

### Step 2: Identify the units of force and velocity
- The unit of force \( F_d \) is Newton (N).
- The unit of velocity \( v \) is meters per second (m/s).

### Step 3: Express the unit of force in terms of base units
We know that:
\[ 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \]
Thus, the unit of force in base units is:
\[ \text{N} = \text{kg} \cdot \text{m/s}^2 \]

### Step 4: Substitute the units into the equation
From the equation \( F_d = C \cdot v \), we can rearrange it to find the unit of \( C \):
\[ C = \frac{F_d}{v} \]
Substituting the units:
\[ C = \frac{\text{N}}{\text{m/s}} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m/s}} \]

### Step 5: Simplify the expression
When we simplify the expression:
\[ C = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m/s}} = \text{kg} \cdot \frac{\text{s}}{\text{s}^2} = \frac{\text{kg}}{\text{s}} \]

### Conclusion
The units of the constant of proportionality \( C \) are:
\[ \text{kg/s} \]

### Final Answer
The correct option for the units of the constant of proportionality is \( \text{kg/s} \).

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