To find the equilibrium of an object, you determine the conditions under which it remains at rest or moves with constant velocity. Based on the provided reference, an object is in static equilibrium if both the sum of the external forces exerted on the object and the sum of the external torques (about any axis) are zero. This means there is no net push or pull, and no net rotational effect.
Conditions for Static Equilibrium
Finding the equilibrium state involves applying two fundamental physical principles, often referred to as Newton's Laws applied to extended bodies.
1. Translational Equilibrium (Sum of Forces is Zero)
This condition ensures the object is not accelerating linearly. In other words, the object will either remain at rest or continue moving at a constant velocity.
- Principle: The vector sum of all external forces acting on the object must be zero ($\Sigma \mathbf{F} = 0$).
- Application: You typically break down forces into their components along perpendicular axes (e.g., x, y, and z). The sum of the force components in each direction must individually equal zero:
- $\Sigma F_x = 0$
- $\Sigma F_y = 0$
- $\Sigma F_z = 0$ (for 3D cases)
2. Rotational Equilibrium (Sum of Torques is Zero)
This condition ensures the object is not angularly accelerating. It will either not be rotating or will be rotating at a constant angular velocity.
- Principle: The sum of all external torques acting on the object about any chosen axis must be zero ($\Sigma \mathbf{\tau} = 0$).
- Application: Torque depends on the force applied and the distance from the pivot point (or axis). For planar motion, you usually consider torques causing rotation in one direction (e.g., counter-clockwise, often positive) and those causing rotation in the opposite direction (e.g., clockwise, often negative). The sum must balance out:
- $\Sigma \tau = 0$ (for 2D cases)
Steps to Find Equilibrium
Finding the equilibrium state, particularly for an unknown quantity like an unknown force or position, involves applying these two conditions mathematically.
- Draw a Free-Body Diagram: Illustrate all external forces acting on the object (gravity, tension, normal force, friction, etc.). Indicate their points of application and directions.
- Choose a Coordinate System: Select appropriate perpendicular axes (e.g., x and y).
- Resolve Forces: Break down all forces into their components along your chosen axes.
- Apply the First Condition (Translational Equilibrium): Write equations setting the sum of the x-components of forces to zero ($\Sigma F_x = 0$) and the sum of the y-components of forces to zero ($\Sigma F_y = 0$).
- Choose a Pivot Point: Select any point as the axis for calculating torques. A smart choice (e.g., a point where unknown forces act) can simplify calculations by eliminating those forces from the torque equation.
- Calculate Torques: For each force, calculate the torque it produces about the chosen pivot point ($\tau = rF\sin\theta$, where $r$ is the distance from the pivot to the force's application point, $F$ is the force magnitude, and $\theta$ is the angle between the position vector $\mathbf{r}$ and the force vector $\mathbf{F}$). Determine the direction of each torque (clockwise or counter-clockwise).
- Apply the Second Condition (Rotational Equilibrium): Write an equation setting the sum of the torques (considering their directions) to zero ($\Sigma \tau = 0$).
- Solve the System of Equations: You will typically have a system of linear equations (from the force and torque conditions). Solve these equations simultaneously to find the unknown quantities, such as unknown forces, tensions, or equilibrium positions.
Example Scenario
Imagine a sign hanging from a beam supported by a cable and a hinge. To find the tension in the cable and the forces at the hinge when the sign is in equilibrium:
- You would draw forces: weight of the sign, weight of the beam, tension in the cable (at an angle), and horizontal/vertical forces at the hinge.
- Apply $\Sigma F_x = 0$ and $\Sigma F_y = 0$ to get two equations involving the forces.
- Apply $\Sigma \tau = 0$ about a chosen point (e.g., the hinge to eliminate its unknown forces from the torque equation) to get a third equation.
- Solve the three equations simultaneously for the unknown tension and hinge force components.
By systematically applying these conditions, you can determine the forces, tensions, or positions required for an object to be in static equilibrium.
