To determine the force [tex]\( F \)[/tex] at the cross-section at a distance [tex]\( l \)[/tex] from the end where [tex]\(F_1\)[/tex] is applied on a homogeneous rod of length [tex]\( L \)[/tex] acted upon by two forces [tex]\( F_1 \)[/tex] and [tex]\( F_2 \)[/tex] which are opposite in direction (with [tex]\( F_1 > F_2 \)[/tex]), we can proceed as follows:
1. Difference in Forces:
- Firstly, determine the difference in forces acting on the two ends of the rod.
- The difference is [tex]\( F_1 - F_2 \)[/tex].
2. Force Distribution:
- The forces are opposing each other causing a varying force distribution along the length of the rod.
- The force at any point along the rod will linearly interpolate between [tex]\( F_1 \)[/tex] at one end and [tex]\( F_2 \)[/tex] at the other end.
3. Location [tex]\( l \)[/tex] from [tex]\( F_1 \)[/tex]:
- We need to determine the force at a cross-section located at a distance [tex]\( l \)[/tex] from the end where [tex]\( F_1 \)[/tex] is applied.
4. Linear Interpolation:
- Since the force varies linearly along the length of the rod, at a distance [tex]\( l \)[/tex] from the end with force [tex]\( F_1 \)[/tex], the force [tex]\( F \)[/tex] can be computed by an interpolation formula.
5. Interpolation Formula:
- The linear interpolation can be described as:
[tex]\[
F = F_2 + \left( \frac{F_1 - F_2}{L} \right) \times l
\][/tex]
Here’s a breakdown of the formula:
- [tex]\( \frac{F_1 - F_2}{L} \)[/tex] represents the rate of change of the force per unit length along the rod.
- Multiplying this rate by [tex]\( l \)[/tex] gives us the difference in force from [tex]\( F_2 \)[/tex] to the point at distance [tex]\( l \)[/tex].
- Adding this difference to [tex]\( F_2 \)[/tex], we get the force at the cross-section distance [tex]\( l \)[/tex] from the end where [tex]\( F_1 \)[/tex] is applied.
Thus, the force [tex]\( F \)[/tex] at the cross-section at distance [tex]\( l \)[/tex] from the end where [tex]\( F_1 \)[/tex] is applied is:
[tex]\[
F = F_2 + \frac{l \times (F_1 - F_2)}{L}
\][/tex]
So, from the provided choices, the appropriate answer corresponds to this formula, which is the force [tex]\( F \)[/tex] at the cross-section:
[tex]\[
\boxed{F = F_2 + \frac{l(F_1 - F_2)}{L}}
\][/tex]
