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 determine how the work done changes when the units of length, mass, and time are each doubled.

### Step-by-Step Solution:

1. **Understand the Definition of Work:**
Work (W) is defined as the product of force (F) and displacement (s). Mathematically, this is expressed as:
\[ W = F \cdot s \]

2. **Determine the Dimensions of Work:**
The dimension of force (F) can be derived from Newton's second law, \( F = m \cdot a \), where \( a \) (acceleration) has dimensions \( L T^{-2} \). Thus, the dimension of force is:
\[ [F] = [M][L][T^{-2}] = M L T^{-2} \]
Therefore, the dimension of work is:
\[ [W] = [F] \cdot [s] = (M L T^{-2}) \cdot [L] = M L^2 T^{-2} \]

3. **Identify the Changes in Units:**
If the units of length (L), mass (M), and time (T) are each doubled, we have:
- New length unit: \( L' = 2L \)
- New mass unit: \( M' = 2M \)
- New time unit: \( T' = 2T \)

4. **Substitute New Units into Work Formula:**
The new work done (W') in terms of the new units can be expressed as:
\[ W' = k \cdot (M') \cdot (L')^2 \cdot (T')^{-2} \]
Substituting the new units into the equation gives:
\[ W' = k \cdot (2M) \cdot (2L)^2 \cdot (2T)^{-2} \]
Simplifying this:
\[ W' = k \cdot (2M) \cdot (4L^2) \cdot \left(\frac{1}{4T^2}\right) \]
\[ W' = k \cdot M \cdot L^2 \cdot T^{-2} \cdot 2 = 2W \]

5. **Conclusion:**
The new unit of work done (W') is \( 2 \) times the old unit of work (W). Therefore, the answer is:
\[ \text{The new unit of work done is } 2 \text{ times the old unit of work.} \]