If unit of length, mass and time each be doubled, the new unit of work done is ________ times the old unit of work

Options

A. 4
B. 6
C. 8
D. 2

Answer

To solve the problem, we need to understand how the units of work change when the units of length, mass, and time are doubled.

### Step-by-Step Solution:

1. **Understand the formula for work done**:
Work done (W) is defined as the product of force (F) and displacement (s):
\[ W = F \cdot s \]

2. **Express force in terms of mass, length, and time**:
The unit of force is given by Newton (N), which can be expressed in terms of base units:
\[ F = m \cdot a = \text{kg} \cdot \text{m/s}^2 \]
where \( m \) is mass in kilograms (kg), \( a \) is acceleration in meters per second squared (m/s²).

3. **Substituting the units into the work formula**:
The unit of work can be expressed as:
\[ W = \text{kg} \cdot \text{m/s}^2 \cdot \text{m} = \text{kg} \cdot \text{m}^2/\text{s}^2 \]

4. **Doubling the units of length, mass, and time**:
If we double the units:
- New mass = \( 2 \, \text{kg} \)
- New length = \( 2 \, \text{m} \)
- New time = \( 2 \, \text{s} \)

5. **Substituting the new units into the work formula**:
Now, substituting the new units into the work formula:
\[ W' = (2 \, \text{kg}) \cdot (2 \, \text{m})^2 / (2 \, \text{s})^2 \]
Simplifying this:
\[ W' = (2 \, \text{kg}) \cdot (4 \, \text{m}^2) / (4 \, \text{s}^2) = \frac{8 \, \text{kg} \cdot \text{m}^2}{4 \, \text{s}^2} \]
\[ W' = 2 \cdot \text{kg} \cdot \text{m}^2/\text{s}^2 \]

6. **Relating the new unit of work to the old unit**:
The new unit of work \( W' \) is:
\[ W' = 2 \cdot W \]
Therefore, the new unit of work done is 2 times the old unit of work done.

### Final Answer:
The new unit of work done is **2 times** the old unit of work.

---