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An object is said to be in equilibrium when it is not changing its state of rest or uniform motion. This means two conditions must be met:
The resultant force is the single equivalent force that would produce the same net displacement as the combination of all forces acting on an object. It's calculated by adding all forces vectorially.
Mathematically, if $\vec{F}_1$, $\vec{F}_2$, ..., $\vec{F}_n$ are the forces acting on an object, then the resultant force $\vec{R}$ is given by:
$$ \vec{R} = \vec{F}_1 + \vec{F}_2 + ... + \vec{F}_n $$The resultant moment is the single equivalent turning effect that would produce the same net rotation as the combination of all moments produced by the forces acting on an object. It's calculated by summing the moments vectorially.
The moment (or torque) of a force is given by:
$$ \text{Moment} = \text{Force} \times \text{Distance} \times \sin(\theta) $$ where:The resultant moment is the vector sum of all individual moments.
Mathematically, if $\vec{F}_i$ is a force at a distance $r_i$ from a pivot point, then the moment $\vec{M}_i$ is given by:
$$ \vec{M}_i = \vec{F}_i \times r_i $$The resultant moment $\vec{M}$ is:
$$ \vec{M} = \vec{M}_1 + \vec{M}_2 + ... + \vec{M}_n $$For an object to be in equilibrium, both of the following conditions must be satisfied:
Therefore, an object is in equilibrium if and only if both the resultant force and the resultant moment are zero.
Understanding the conditions for equilibrium – no resultant force and no resultant moment – is fundamental to analyzing the forces acting on objects and predicting their motion. This concept is crucial for solving problems involving levers, hinges, and other mechanical systems.