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

Options

A. `kgs^(-1)`
B. `kgs`
C. `kgms^(-1)`
D. `kgms^(-2)`

Explanation

To determine the units of the constant of proportionality in the damping force equation, we can follow these steps:

### Step 1: Understand the relationship
The damping force \( F \) is directly proportional to the velocity \( V \). This can be expressed mathematically as:
\[ F \propto V \]

### Step 2: Introduce the proportionality constant
To convert the proportionality into an equation, we introduce a constant of proportionality \( k \):
\[ F = k \cdot V \]

### Step 3: Rearrange the equation to find \( k \)
From the equation, we can solve for \( k \):
\[ k = \frac{F}{V} \]

### Step 4: Identify the units of force and velocity
- The unit of force \( F \) in the International System of Units (SI) is Newton (N), which can be expressed as:
\[ 1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2 \]

- The unit of velocity \( V \) is meters per second (m/s).

### Step 5: Substitute the units into the equation for \( k \)
Now we can substitute the units of force and velocity into the equation for \( k \):
\[ \text{Units of } k = \frac{\text{Units of } F}{\text{Units of } V} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m/s}} \]

### Step 6: Simplify the units
When we simplify the units:
\[ \text{Units of } k = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m/s}} = \frac{\text{kg} \cdot \text{m}}{\text{s}^2} \cdot \frac{\text{s}}{\text{m}} = \frac{\text{kg}}{\text{s}} \]

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

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