If the unit of length , mass and time each be doubled , the unit of work is increased by

Options

A. 5 times
B. 2 times
C. 3 times
D. 4 times

Answer

To solve the problem of how the unit of work changes when the units of length, mass, and time are each 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 (s):
\[ W = F \cdot s \]

### Step 2: Express force in terms of mass and acceleration
Force can be expressed using Newton's second law:
\[ F = m \cdot a \]
where \( m \) is mass and \( a \) is acceleration.

### Step 3: Relate acceleration to displacement and time
Acceleration can be expressed as:
\[ a = \frac{v}{t} \]
where \( v \) is velocity and \( t \) is time.

### Step 4: Substitute acceleration into the work formula
Substituting for force in the work formula, we get:
\[ W = (m \cdot a) \cdot s = (m \cdot \frac{v}{t}) \cdot s \]
This simplifies to:
\[ W = \frac{m \cdot v \cdot s}{t} \]

### Step 5: Identify the initial units of work
Let’s denote the initial units of length, mass, and time as:
- Length = \( l \)
- Mass = \( m \)
- Time = \( t \)

Thus, the initial work done can be expressed as:
\[ W_{initial} = \frac{m \cdot v \cdot l}{t} \]

### Step 6: Analyze the changes when units are doubled
If the units of length, mass, and time are doubled:
- New length = \( 2l \)
- New mass = \( 2m \)
- New time = \( 2t \)

### Step 7: Substitute the new units into the work formula
The new work done can be expressed as:
\[ W_{new} = \frac{(2m) \cdot v \cdot (2l)}{2t} \]

### Step 8: Simplify the new work expression
Now simplifying the new work expression:
\[ W_{new} = \frac{(2m) \cdot v \cdot (2l)}{2t} = \frac{4m \cdot v \cdot l}{2t} = 2 \cdot \frac{m \cdot v \cdot l}{t} \]
Thus:
\[ W_{new} = 2 \cdot W_{initial} \]

### Conclusion
The unit of work is increased by a factor of 2 when the units of length, mass, and time are each doubled.

### Final Answer
The unit of work is increased by **2 times**.

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