To determine the conditions under which a body under the action of five forces can be in equilibrium, we need to analyze the forces acting on the body in the x-y plane.
### Step-by-Step Solution:
1. **Understanding Equilibrium**:
A body is said to be in equilibrium when the net force acting on it is zero. This means that the vector sum of all forces acting on the body must equal zero.
2. **Resolving Forces into Components**:
Any force can be resolved into its components along the x-axis and y-axis. For a force \( F \), the x-component can be represented as \( F_x = F \cos(\theta) \) and the y-component as \( F_y = F \sin(\theta) \), where \( \theta \) is the angle the force makes with the x-axis.
3. **Applying the Condition for Equilibrium**:
For the body to be in equilibrium under the action of five forces, the sum of the x-components of all the forces must be zero:
\[
\sum F_x = 0
\]
Similarly, the sum of the y-components must also be zero:
\[
\sum F_y = 0
\]
4. **Setting Up the Equations**:
Let’s denote the five forces as \( F_1, F_2, F_3, F_4, F_5 \). The x and y components of these forces can be expressed as:
\[
F_{1x}, F_{2x}, F_{3x}, F_{4x}, F_{5x} \quad \text{(x-components)}
\]
\[
F_{1y}, F_{2y}, F_{3y}, F_{4y}, F_{5y} \quad \text{(y-components)}
\]
The conditions for equilibrium can then be written as:
\[
F_{1x} + F_{2x} + F_{3x} + F_{4x} + F_{5x} = 0
\]
\[
F_{1y} + F_{2y} + F_{3y} + F_{4y} + F_{5y} = 0
\]
5. **Conclusion**:
Therefore, for a body under the action of five forces to be in equilibrium, the sum of the resolved components along the x-axis must be zero, and the sum of the resolved components along the y-axis must also be zero. This leads us to the conclusion that both conditions must be satisfied simultaneously.
### Final Answer:
A body under the action of five forces can be in equilibrium only if:
- The sum of the resolved components along the x-axis is zero.
- The sum of the resolved components along the y-axis is zero.
