Equilibrium is the state of a particle when it experiences no net force or torque, allowing it to maintain a constant velocity or remain at rest. Equilibrium focuses on the conditions under which a particle achieves the balance.
Definition of Equilibrium
Equilibrium of a particle is a state in which the net external force and net external torque acting on the particle are zero. This can be mathematically expressed as:
F = 0 (Translational equilibrium)
τ = 0 (Rotational equilibrium)
Rotational equilibrium is more relevant to rigid bodies, translational equilibrium is vital for particles, as they do not have dimensions to consider torque.
Concurrent Forces:– Forces acting together on a body at the same point are called concurrent forces. A number of concurrent forces acting on a body are said to be in equilibrium, if the resultant of these forces is zero or if the concurrent forces can be represented completely by the sides of a polygon taken in the same order.
Mathematically : F1 + F2 + F3 +……Fn = 0
Types of Equilibrium
1.Static Equilibrium:
A particle is said to be in static equilibrium when it is at rest and remains static under the action of forces.
Example: A book lying on a table is in static equilibrium because the upward normal force balances its downward weight.
2. Dynamic Equilibrium:
A particle is in dynamic equilibrium when it is moving with a constant velocity. Here, the net force is zero, but the particle maintains motion.
Example: A car travel at a constant speed on a straight road without acceleration.
Conditions for Equilibrium
To achieve equilibrium, the following conditions must be satisfied:
1.First Condition: Zero Net Force
For a particle to be in equilibrium, the vector sum of all forces acting on it must be zero: F = 0
This ensures that the particle does not accelerate and either stays at rest or moves uniformly.
2. Second Condition: Zero Net Torque (If Applicable)
Torque is primarily associated with rigid bodies, for a particle in equilibrium, there is no significant torque acting, as it lacks dimensions to create a moment arm.
Equilibrium in Two Dimensions
For equilibrium in two dimensions, the forces acting on a particle can be resolved into their components along the x- and y-axes. The equilibrium conditions are:
Fx = 0
Fy = 0
This means that the horizontal and vertical forces must independently balance each other.
Example: Suppose a particle suspended by two strings. The tension in the strings (T1 and T2) and the weight of the particle (W) form a system of forces. Resolving forces:
Horizontally: T1cosθ1 = T2cosθ2
Vertically: T1sinθ1 + T2sinθ2 = W
Equilibrium in Three Dimensions
In three dimensions, the forces must balance along the x-, y-, and z-axes:
Fx = 0
Fy = 0
Fz = 0
Such problems often is in vectors and require understanding their components in space
Applications of Equilibrium
Engineering Structures:
Bridges, buildings, and cranes rely on equilibrium principles to ensure stability.
Mechanics:
Equilibrium is critical in analysing forces in systems such as pulleys and levers.
Daily Life:
Objects placed on flat surfaces are often in static equilibrium, ensuring they remain stable
Methods to Analyse Equilibrium
Free-Body Diagrams (FBDs):
An FBD represents all forces acting on a particle, aiding in visualising and solving equilibrium problems.
Mathematical Analysis:
Using equations of force components to verify balance.
Vector Approach:
Summing up vectors to ensure the net force equals zero.
Significance in Physics
The concept of equilibrium helps simplify complex problems by focusing on balance. It serves as to know about stability, motion, and mechanics in a wide range of physical systems.
Note:
The equilibrium of a particle is an essential concept for the balance of forces. Whether static or dynamic, equilibrium conditions helps to predict the behaviour of particles in various scenarios, from resting objects to moving systems.
Equilibrium of a particle occurs when the net external force acting on it is zero (F⃗ = 0), ensuring that the particle remains at rest (static equilibrium) or moves with constant velocity (dynamic equilibrium).
The two main types of equilibrium are:
Static Equilibrium: The particle is at rest, e.g., a book lying on a table.
Dynamic Equilibrium: The particle is in motion with constant velocity, e.g., a car moving uniformly on a straight road.
The conditions are:
The vector sum of all forces acting on the particle must be zero: F = 0.
In two dimensions, this means Fx = 0 and Fy = =0. In three dimensions, Fx = 0, Fy = 0, and Fz = 0.
A free-body diagram (FBD) is a visual representation of all forces acting on a particle. By resolving these forces into components and applying the conditions of equilibrium (Fx = 0 and Fy = 0), the system can be analysed to find unknown forces or verify balance.
Static Equilibrium: The particle is at rest, with zero velocity and acceleration.
Dynamic Equilibrium: The particle is in motion with constant velocity, meaning acceleration is zero but velocity is non-zero.
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