A physical quantity is measured and the result is expressed as nu where u is the unit used and n is the numberical value. If the result is expressed in various units then

Options

A. `n prop` size of `u`
B. `n prop u^2`
C. `n prop sqrtu`
D. `n prop (1)/(u).`

Answer

To solve the problem, we need to understand the relationship between the numerical value (N) and the unit (U) when expressing a physical quantity. Let's break down the steps:

### Step 1: Understand the relationship between N and U
When a physical quantity is expressed as \( N \cdot U \), where \( N \) is the numerical value and \( U \) is the unit, changing the unit will affect the numerical value.

### Step 2: Example with meters and centimeters
Let's take an example where we have a physical quantity measured as 5000 meters. Here, \( N = 5000 \) and \( U = \text{meters} \).

### Step 3: Convert meters to centimeters
To convert meters to centimeters, we know that:
1 meter = 100 centimeters.

Therefore, to convert 5000 meters to centimeters:
\[ 5000 \text{ meters} = 5000 \times 100 \text{ centimeters} = 500000 \text{ centimeters} \]
Now, in this case, \( N \) becomes 500000 and \( U \) is now centimeters.

### Step 4: Convert meters to kilometers
Now, let’s convert the same physical quantity into kilometers:
1 kilometer = 1000 meters.

Thus, to convert 5000 meters to kilometers:
\[ 5000 \text{ meters} = \frac{5000}{1000} \text{ kilometers} = 5 \text{ kilometers} \]
Here, \( N = 5 \) and \( U = \text{kilometers} \).

### Step 5: Analyze the relationship
From the examples:
- When the unit changed from meters to centimeters, \( N \) increased from 5000 to 500000.
- When the unit changed from meters to kilometers, \( N \) decreased from 5000 to 5.

This indicates that as the unit \( U \) increases (from meters to kilometers), the numerical value \( N \) decreases, and vice versa.

### Step 6: Conclusion on the relationship
This leads us to conclude that \( N \) is inversely proportional to \( U \):
\[ N \propto \frac{1}{U} \]

Thus, if you express a physical quantity in different units, the numerical value will change inversely with the unit used.

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