The magnitude of force is `100 N`. What will be its value if the units of mass and time are doubled and that of length is halved?

Options

A. `25`
B. `100`
C. `200`
D. `400`

Explanation

To solve the problem, we need to analyze how the changes in the units of mass, time, and length affect the value of force. The force is given by the formula:

\[ F = m \cdot a \]

Where:
- \( F \) is the force,
- \( m \) is the mass,
- \( a \) is the acceleration.

The unit of force in the SI system is Newton (N), which can be expressed in terms of base units as:

\[ 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \]

This can be represented in dimensional form as:

\[ [F] = [M][L][T^{-2}] \]

Where:
- \( [M] \) is the dimension of mass,
- \( [L] \) is the dimension of length,
- \( [T] \) is the dimension of time.

Given:
- The original force \( F = 100 \, \text{N} \).
- The units of mass and time are doubled, and the unit of length is halved.

### Step-by-step Solution:

1. **Identify the original dimensions of force:**
\[ F = m \cdot a = m \cdot \frac{L}{T^2} \]
Here, \( a = \frac{L}{T^2} \).

2. **Substituting the original units into the force equation:**
\[ F = m \cdot \frac{L}{T^2} \]
In SI units, this becomes:
\[ F = 1 \, \text{kg} \cdot \frac{1 \, \text{m}}{(1 \, \text{s})^2} = 1 \, \text{N} \]

3. **Adjust the units according to the problem statement:**
- Mass is doubled: \( m' = 2m \)
- Length is halved: \( L' = \frac{1}{2}L \)
- Time is doubled: \( T' = 2T \)

4. **Substituting the new units into the force equation:**
\[ F' = m' \cdot \frac{L'}{(T')^2} \]
Substituting the new values:
\[ F' = (2m) \cdot \frac{\frac{1}{2}L}{(2T)^2} \]

5. **Simplifying the equation:**
\[ F' = (2m) \cdot \frac{\frac{1}{2}L}{4T^2} \]
\[ F' = \frac{2m \cdot \frac{1}{2}L}{4T^2} = \frac{mL}{4T^2} \]

6. **Relating the new force to the original force:**
Since \( F = m \cdot \frac{L}{T^2} \), we can express \( F' \) in terms of \( F \):
\[ F' = \frac{1}{4}F \]

7. **Calculating the new force:**
Given \( F = 100 \, \text{N} \):
\[ F' = \frac{1}{4} \cdot 100 \, \text{N} = 25 \, \text{N} \]

### Final Answer:
The new value of the force is \( 25 \, \text{N} \).