If the unit of force and length be ecah increased by four times, then the unit of energy is increased by

Options

A. 16 times
B. 8 times
C. 2 times
D. 4 times

Explanation

To solve the problem, we need to understand how the units of force, length, and energy are related.

1. **Understanding Units**:
- The unit of force (F) is given by Newton (N), which is defined as \(1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2\).
- The unit of length (L) is meter (m).
- The unit of energy (E) is Joule (J), which is defined as \(1 \, \text{J} = 1 \, \text{N} \cdot \text{m} = 1 \, \text{kg} \cdot \text{m}^2/\text{s}^2\).

2. **Initial Units**:
- The initial unit of energy can be expressed as:
\[ E = F \cdot L = (1 \, \text{N}) \cdot (1 \, \text{m}) = (1 \, \text{kg} \cdot \text{m/s}^2) \cdot (1 \, \text{m}) = 1 \, \text{kg} \cdot \text{m}^2/\text{s}^2 \]

3. **Increasing Units**:
- If the unit of force is increased by 4 times, the new unit of force becomes:
\[ F' = 4 \cdot F = 4 \cdot (1 \, \text{N}) = 4 \, \text{N} \]
- If the unit of length is also increased by 4 times, the new unit of length becomes:
\[ L' = 4 \cdot L = 4 \cdot (1 \, \text{m}) = 4 \, \text{m} \]

4. **Calculating New Energy**:
- The new unit of energy can be calculated using the new units of force and length:
\[ E' = F' \cdot L' = (4 \, \text{N}) \cdot (4 \, \text{m}) = 16 \, \text{N} \cdot \text{m} \]
- Since \(1 \, \text{N} \cdot \text{m} = 1 \, \text{J}\), we have:
\[ E' = 16 \, \text{J} \]

5. **Comparing Energies**:
- The original unit of energy was \(1 \, \text{J}\).
- The new unit of energy is \(16 \, \text{J}\).
- Therefore, the increase in the unit of energy is:
\[ \text{Increase} = \frac{E'}{E} = \frac{16 \, \text{J}}{1 \, \text{J}} = 16 \]

Thus, if the unit of force and length is each increased by four times, then the unit of energy is increased by **16 times**.