Rigid Bodies in Equilibrium

Rigid Bodies in Equilibrium

Introduction to Rigid Bodies

  • A rigid body is an idealised concept where the body does not deform under the application of force.
  • Equilibrium refers to a state where the resultant force and torque acting on a rigid body are zero.
  • If a body is in equilibrium, it means that it is either at rest or moving with a constant velocity.

Conditions for Equilibrium

  • There are two conditions required for a rigid body to be in equilibrium:
    • The vector sum of all forces acting on the body must be zero - translational equilibrium.
    • The sum of all torques, or moments, about any point must also be zero - rotational equilibrium.

Centre of Mass

  • The centre of mass is a point in a rigid body where the total weight of the body can be assumed to act.
  • For symmetric bodies, the centre of mass is at the geometric centre.
  • For an irregular shape or system of particles, the position of the centre of mass can be calculated using the formula:
    • ∑miri / ∑mi
    • where ‘mi’ represents the mass of the i-th object and ‘ri’ represents its distance from the reference point.

Resultant Force

  • A single force that has the same effect as all the forces acting on the body is termed as the resultant force.
  • The resultant force is found by vector addition of all the forces acting on the body.

Torque of a Force

  • The torque, or the moment of a force about a point, is a measure of the rotational effect of the force about the point.
  • The torque τ is calculated by multiplying the force ‘f’ by the perpendicular distance ‘r’ from the point to the line of action of the force.
    • τ = r f sin(θ)

Principles of Moments

  • According to the principle of moments, for a body to be in equilibrium, the sum of clockwise moments about any point needs to be equal to the sum of anticlockwise moments about that same point.

Free Body Diagrams

  • Free body diagrams are visual representations used to analyze the forces and torques acting on a rigid body.
  • These diagrams include all the forces acting on the body, as well as their direction and point of application.

Solving Equilibrium Problems

  • Step 1: Draw a free body diagram that represents the problem.
  • Step 2: Identify all the forces acting on the body and write down the equations of equilibrium.
  • Step 3: Use the equations of equilibrium to solve for the forces or moments, as required.
  • Use the principles of vector addition and principles of moments to solve these problems.