If the units of length, mass and time are doubled, then the new unit of work expressed in terms of L, M and T

Options

A. becomes four times
B. becomes two times
C. becomes eight times
D. is not changed

Explanation

To solve the problem of how the unit of work changes when the units of length, mass, and time are doubled, we can follow these steps:

### Step 1: Understand the formula for work
Work (W) is defined as the product of force (F) and displacement (d). Mathematically, this is expressed as:
\[ W = F \cdot d \]

### Step 2: Define force in terms of mass and acceleration
Force can be defined using Newton's second law:
\[ F = m \cdot a \]
Where:
- \( m \) is mass
- \( a \) is acceleration

### Step 3: Express acceleration in terms of length and time
Acceleration (a) can be expressed as:
\[ a = \frac{L}{T^2} \]
Where:
- \( L \) is length
- \( T \) is time

### Step 4: Substitute the expression for force into the work formula
Substituting for force in the work equation, we get:
\[ W = m \cdot a \cdot d = m \cdot \left(\frac{L}{T^2}\right) \cdot d \]

### Step 5: Substitute displacement with length
Assuming displacement (d) is also a length, we can write:
\[ W = m \cdot \left(\frac{L}{T^2}\right) \cdot L = m \cdot \frac{L^2}{T^2} \]

### Step 6: Write the unit of work in terms of L, M, and T
Now we can express the unit of work in terms of dimensions:
\[ W = M \cdot \frac{L^2}{T^2} \]
Thus, the dimensional formula for work is:
\[ [W] = M L^2 T^{-2} \]

### Step 7: Analyze the effect of doubling the units
If the units of length (L), mass (M), and time (T) are doubled:
- New mass \( M' = 2M \)
- New length \( L' = 2L \)
- New time \( T' = 2T \)

### Step 8: Substitute the new units into the work formula
Now substituting these new units into the work formula:
\[ W' = (2M) \cdot \frac{(2L)^2}{(2T)^2} \]
\[ W' = (2M) \cdot \frac{4L^2}{4T^2} \]
\[ W' = 2M \cdot \frac{L^2}{T^2} \]
\[ W' = 2 \cdot (M \cdot \frac{L^2}{T^2}) \]
Thus, the new unit of work becomes:
\[ W' = 2W \]

### Conclusion
The new unit of work, when the units of length, mass, and time are doubled, is 2 times the original unit of work.