The necessary condition of equilibrium of a body is-
∑Fx = 0, ∑Εy = 0, ∑Μ = 0
Understanding Equilibrium Conditions for a Body
Equilibrium is a state where a body remains at rest or continues to move with a constant velocity. For a body to be in complete equilibrium, two conditions must be satisfied simultaneously:
- The net external force acting on the body must be zero.
- The net external torque (or moment) acting on the body about any point must be zero.
Analyzing the Conditions for Equilibrium
Let's break down these conditions:
- Translational Equilibrium: This means the body is not accelerating linearly. The vector sum of all forces acting on the body is zero ($\sum \vec{F} = 0$). In a 2D plane (like the xy-plane), this can be broken down into components:
- Sum of forces in the x-direction is zero ($\sum F_x = 0$).
- Sum of forces in the y-direction is zero ($\sum F_y = 0$).
- Rotational Equilibrium: This means the body is not angularly accelerating. The vector sum of all moments (or torques) acting on the body about any point is zero ($\sum \vec{\tau} = 0$ or $\sum \vec{\Mu} = 0$). In 2D, this typically means the sum of moments about a single point in the plane is zero ($\sum \Mu = 0$).
Evaluating the Given Options
Let's look at the provided options in light of these conditions:
- Option 1: $\sum F_y = 0, \sum \Mu = 0$. This only considers forces in the y-direction and moments. It ignores forces in the x-direction (and z-direction if applicable), which are necessary for complete translational equilibrium.
- Option 2: $\sum F_x = 0, \sum \Mu = 0$. This only considers forces in the x-direction and moments. It ignores forces in the y-direction (and z-direction if applicable), which are necessary for complete translational equilibrium.
- Option 3: $\sum F_x = 0, \sum F_y = 0$. This satisfies the condition for translational equilibrium in 2D. However, it does not include the condition for rotational equilibrium ($\sum \Mu = 0$). A body can have zero net force but still rotate if there is a net moment (e.g., a couple).
- Option 4: $\sum F_x = 0, \sum \Epsilon_y = 0, \sum \Mu = 0$. This option includes $\sum F_x = 0$ for translational equilibrium in the x-direction and $\sum \Mu = 0$ for rotational equilibrium. The term $\sum \Epsilon_y = 0$ appears to be a typo and likely intended to represent $\sum F_y = 0$, which is the condition for translational equilibrium in the y-direction. Assuming this interpretation, this option includes conditions for translational equilibrium in both x and y directions (in 2D) and rotational equilibrium. Therefore, it lists the necessary conditions for a body to be in complete equilibrium in a plane.
Based on the standard conditions for equilibrium, a body in 2D is in equilibrium if $\sum F_x = 0$, $\sum F_y = 0$, and $\sum \Mu = 0$. Option 4, despite the likely typo, includes the necessary components for describing complete equilibrium in a plane: force balance along one axis ($\sum F_x = 0$), force balance along another axis (intended as $\sum F_y = 0$, represented by $\sum \Epsilon_y = 0$), and moment balance ($\sum \Mu = 0$).
Conclusion on Necessary Equilibrium Conditions
For a body to be in complete equilibrium, it must satisfy both translational and rotational equilibrium. This means the net force in all directions must be zero, and the net moment about any point must be zero. Option 4 correctly identifies the combination of force balance conditions (in x and likely y directions) and moment balance as necessary for equilibrium.
| Equilibrium Type | Condition | Explanation |
|---|---|---|
| Translational | $\sum \vec{F} = 0$ | Net force is zero; no linear acceleration. |
| Rotational | $\sum \vec{\Mu} = 0$ | Net moment (torque) is zero; no angular acceleration. |
Revision Table: Key Equilibrium Concepts
| Concept | Definition | Conditions |
|---|---|---|
| Static Equilibrium | Body is at rest and remains at rest. | $\sum \vec{F} = 0$ and $\sum \vec{\Mu} = 0$ |
| Dynamic Equilibrium | Body moves with constant velocity (zero acceleration). | $\sum \vec{F} = 0$ and $\sum \vec{\Mu} = 0$ |
| Force Balance | Sum of external forces is zero. | $\sum F_x = 0, \sum F_y = 0$ (in 2D) |
| Moment Balance | Sum of external moments (torques) is zero. | $\sum \Mu = 0$ (in 2D, about any point) |
Additional Information: Equilibrium and Free-Body Diagrams
To analyze the equilibrium of a body, it is essential to draw a free-body diagram (FBD). An FBD is a diagram that isolates the body and shows all external forces and moments acting on it. This helps in writing the correct equilibrium equations ($\sum F_x = 0$, $\sum F_y = 0$, $\sum \Mu = 0$).
Different types of supports (like pins, rollers, fixed supports) exert different types of reaction forces and moments, which must be included in the FBD when solving problems related to the equilibrium of structures or rigid bodies.
The choice of the point about which to take moments is arbitrary; if a body is in rotational equilibrium, the sum of moments about any point will be zero. Choosing a point where several forces intersect can simplify calculations, as those forces will not create a moment about that point.
Important Questions from Equilibrium and Friction
How does a lubricant reduce friction between moving parts of a machine?
The forces whose line of action lie along the same line are known as:
The forces which meet at one point and have their line of action in different planes are called
If in a planar system, only 2 reaction forces are acting, then the system is:-
By applying the static equations i.e. ∑H = 0, ∑V = 0 and ∑M = 0. To a determine structure, we may determine:
