physics

Equilibrium of Forces

• Definition of Equilibrium
    - A body is said to be in equilibrium when there is no net force acting on it.
    - For a body to be in equilibrium:
      - It either remains at rest or moves in a straight line with uniform velocity.
      - It is either not rotating at all or rotating at a constant angular velocity.

• Types of Equilibrium
    - Static Equilibrium
      - Definition: There is no net force on the body; it is at rest.
    - Dynamic Equilibrium
      - Definition: The body moves with uniform velocity along a straight line or rotates with uniform angular velocity about a fixed axis through its center of mass.

Resultant and Equilibrant Forces

• Resultant Force
    - Definition: A single force that has the same effect as two or more forces added together, both in magnitude and direction.
    - Example:
      - Given: Two forces of 3N and 4N parallel to each other, acting in the same direction.
      - Calculation:
        - Resultant force = 3N + 4N = 7N
      - Result: This 7N force represents both the 3N and 4N forces in magnitude and direction.

• Equilibrant Force
    - Definition: A single force which balances all other forces taken together; it is equal to the resultant force but in the opposite direction.
    - Representation:
      - Let R represent the resultant force, F1 and F2 represent other forces, then F3 is the equilibrant of F1 and F2.

Equilibrium of Three Forces Acting at a Point

• Triangle of Forces Principle
    - If three forces act at equilibrium, they can be represented in magnitude and direction by the three sides of a triangle taken in order.

Moment of a Force

• Definition:
    - The moment of a force about a point is the turning effect produced by the force around that point.
    - It is equal to the product of the force and the perpendicular distance from the point to the line of action of the force.

• Moment of Force (F) About Point P:
    - For a force inclined at an angle, it can be expressed as:
      - $ext{Moment} = F imes d$
    - Here, if the force is inclined, the perpendicular distance (d) can be calculated as:
      - $d = x imes ext{sin}( heta)$
    - Therefore, the moment of force F about point P is:
      - $ext{Moment} = F imes x imes ext{sin}( heta)$

Conditions of Equilibrium of a Body Under the Action of Parallel Coplanar Forces

1. The algebraic sum of forces acting on the body in any given direction must equal zero.

2. The algebraic sum of the moments of all forces about any point on the body must equal zero.
       - Total clockwise moment of the forces about any point must equal the total anticlockwise moments about the same point.

• Example:
    - A uniform bar AB is balanced on a knife edge which is 60 cm from point B. A mass of 22 g hangs at point C, which is 10 cm from point A.
    - Objective: Calculate the mass of the bar.

Couple

• Definition:
    - A couple is a system of two equal, parallel, but oppositely directed forces not acting in a straight line.
    - The two forces have zero resultant, but they have a turning effect on the body upon which they act.

• Applications of Couple:
    - Turning a tap with our fingers.
    - Turning the steering wheel of a car.

• Moment of a Couple:
    - Definition: The product of one of the forces and the perpendicular distance between the lines of action of the two forces.
    - The distance between the two equal forces is referred to as the arm of the couple.
    - The moment of a couple is called torque and is also referred to as the magnitude of the couple.

Conditions of Equilibrium under the Action of Non-Parallel Coplanar Forces

1. The vector sum of all forces acting on the body must equal zero.

2. The algebraic sum of the moments of all forces about any axis perpendicular to the plane of the forces must equal zero.

Centre of Gravity (G)

• Definition:
    - The center of gravity of a body is the point through which the line of action of the weight of the body always passes.
    - It is the point at which the entire weight of the body appears to be concentrated.
      

• Location of the Center of Gravity:
    - For a uniform rod: at its midpoint.
    - For a uniform circular plate: at its center.
    - For a uniform square or rectangular sheet: at the point of intersection of its diagonals.
    - For a uniform triangular plate: at the point of intersection of the medians.

Stability of Objects

• Types of Equilibrium:
    - Stable Equilibrium:
      - Definition: A body is said to be in stable equilibrium if, when slightly displaced, it tends to return to its original position.
      - Example: A cone resting on its base, a ball or sphere in the middle of a bowl.
    - Unstable Equilibrium:
      - Definition: A body is said to be in unstable equilibrium if, when slightly displaced, it tends to move further away from its original position.
      - Example: A cone resting on its apex, a ball or sphere resting on an inverted bowl, a tight-rope walker.
    - Neutral Equilibrium:
      - Definition: A body is in neutral equilibrium if, when slightly displaced, it tends to come to rest in its new position.
      - Example: A cone resting on its curved surface, a cylinder resting on its curved surface, a ball resting on a smooth horizontal table.