To determine the length of the spring when a 100N force is applied, we can use Hooke's Law. Hooke's Law states that the force [tex]\( F \)[/tex] applied to a spring is directly proportional to the extension [tex]\( x \)[/tex] of the spring, given by:
[tex]\[ F = k \cdot x \][/tex]
where:
- [tex]\( F \)[/tex] is the force applied,
- [tex]\( x \)[/tex] is the extension of the spring,
- [tex]\( k \)[/tex] is the spring constant.
First, let's determine the spring constant [tex]\( k \)[/tex] using the initial conditions:
- Initial force [tex]\( F_1 \)[/tex] = 80N
- Initial extension [tex]\( x_1 \)[/tex] = 0.4m
[tex]\[ k = \frac{F_1}{x_1} = \frac{80N}{0.4m} = 200 \, \text{N/m} \][/tex]
Next, we apply the new force [tex]\( F_2 \)[/tex] = 100N and calculate the new extension [tex]\( x_2 \)[/tex]:
[tex]\[ x_2 = \frac{F_2}{k} = \frac{100N}{200 \, \text{N/m}} = 0.5m \][/tex]
The initial natural length of the spring is given as 8m, so the total length of the spring when the 100N force is applied will be:
[tex]\[ \text{Total length of the spring} = \text{Natural length} + \text{Extension} \][/tex]
[tex]\[ \text{Total length of the spring} = 8m + 0.5m = 8.5m \][/tex]
The provided options seem to fall in a different context, which means there might be a mistake in the options provided as the calculated length does not directly match any of them. However, if we assume the question refers to the extension only, then the option that matches the nearest logic here:
[tex]\(\boxed{0.5m}\)[/tex]
Hence, considering the natural length wasn't asked within the options, it ideally matches correctly with the option:
[tex]\[
\boxed{0.5m \text{ Extension length such option who considers as the right answer}}
\][/tex]
