A unitless quantity

Options

A. never has a nonzero dimension
B. always has a nonzero dimension
C. may have a nonzero dimension.
D. does not exist

Answer

To solve the question regarding a unitless quantity, let's break down the concepts step by step.

### Step-by-Step Solution:

1. **Understanding Unitless Quantities**:
- A unitless quantity is defined as a quantity that does not have any associated units. This means it is a pure number or ratio.

2. **Examples of Unitless Quantities**:
- A common example of a unitless quantity is strain. Strain is defined as the ratio of the change in length to the original length:
\[ \text{Strain} = \frac{\text{Change in Length}}{\text{Original Length}} \]
- Both the change in length and the original length have dimensions of length (L), so when you divide them, the units cancel out:
\[ \text{Strain} = \frac{L}{L} = 1 \quad (\text{unitless}) \]

3. **Dimensional Analysis**:
- When performing dimensional analysis on strain, we see that the dimensions cancel out:
\[ \text{Dimension of Strain} = \frac{[L]}{[L]} = [L]^0 = 1 \]
- This indicates that strain has no dimensions, confirming it is a unitless quantity.

4. **Evaluating the Given Options**:
- The options provided are:
- A) Never has a non-zero dimension
- B) Always has a non-zero dimension
- C) May have a non-zero dimension
- D) Does not exist
- From our analysis, we can conclude:
- A) is correct because unitless quantities always have zero dimensions.
- B) is incorrect because unitless quantities do not have non-zero dimensions.
- C) is incorrect because unitless quantities cannot have non-zero dimensions.
- D) is incorrect because unitless quantities do exist (e.g., strain).

5. **Conclusion**:
- The correct answer is **A) Never has a non-zero dimension**.