To solve the problem, we need to find the numerical value of the force in the new system of units after the specified changes to the units of mass, length, and time.
### Step-by-Step Solution:
1. **Identify the Given Information:**
- Initial force \( F = 5 \, \text{N} \)
- 1 Newton \( = 1 \, \text{kg} \cdot \text{m/s}^2 \)
- New unit of mass \( = \frac{1}{2} \, \text{kg} \)
- New unit of length \( = 2 \, \text{m} \)
- New unit of time \( = 2 \, \text{s} \)
2. **Convert the Force into Base Units:**
- The force in base units is:
\[
F = 5 \, \text{N} = 5 \, \text{kg} \cdot \text{m/s}^2
\]
3. **Express the New Units:**
- In the new system:
- Mass \( m' = \frac{1}{2} \, \text{kg} \)
- Length \( l' = 2 \, \text{m} \)
- Time \( t' = 2 \, \text{s} \)
4. **Substitute the New Units into the Force Equation:**
- The formula for force is:
\[
F = m \cdot a
\]
- Where acceleration \( a \) can be expressed as:
\[
a = \frac{\text{change in velocity}}{\text{time}} = \frac{\text{length}}{\text{time}^2}
\]
- In the new units, the acceleration becomes:
\[
a' = \frac{l'}{(t')^2} = \frac{2 \, \text{m}}{(2 \, \text{s})^2} = \frac{2 \, \text{m}}{4 \, \text{s}^2} = \frac{1}{2} \, \text{m/s}^2
\]
5. **Calculate the New Force:**
- Now substitute the new mass and acceleration into the force equation:
\[
F' = m' \cdot a' = \left(\frac{1}{2} \, \text{kg}\right) \cdot \left(\frac{1}{2} \, \text{m/s}^2\right) = \frac{1}{4} \, \text{kg} \cdot \text{m/s}^2
\]
6. **Final Result:**
- Since \( 1 \, \text{kg} \cdot \text{m/s}^2 = 1 \, \text{N} \), we can express the new force in Newtons:
\[
F' = \frac{1}{4} \, \text{N}
\]
- Therefore, the numerical value of the force in the new system of units is:
\[
F' = \frac{5}{4} \, \text{N}
\]
### Final Answer:
The numerical value of the force in the new system of units is \( \frac{5}{4} \, \text{N} \).
