Equilibrium Of A Rigid Bodymoments And Center Of Gravity Topic

Equilibrium of a Rigid Body - Moments and Center of Gravity

1. Moments and Torque:

  • Moment of a force:
  • Defined as the product of the force and the perpendicular distance from the point of rotation (fulcrum) to the line of action of the force.
  • SI unit: Newton-meter (N-m)
  • Torque:
  • Defined as the rotational effect of a force.
  • Mathematically, torque (τ) = F × r × sinθ
    • F = magnitude of the force
    • r = perpendicular distance from the point of rotation to the line of action of the force
    • θ = angle between the force and the lever arm (perpendicular distance)
  • Relationship between moment of a force and torque:
  • Torque is the moment of a force about a specific point.

2. Center of Gravity:

  • Center of gravity:
    • The point at which the entire weight of an object can be considered to be concentrated.
    • For symmetrical objects, the center of gravity is usually at the geometric center.
    • For irregularly shaped objects, the center of gravity can be determined by various methods, including:
      • Suspension method: Suspend the object from a point and draw a vertical line through the point of suspension. Repeat the process for another point of suspension, and the intersection of the two lines gives the center of gravity.
      • Balancing method: Place the object on a horizontal surface and find the point at which it balances in all directions. This point is the center of gravity.
  • Properties of the center of gravity:
  • The center of gravity of an object is fixed and does not change with its orientation.
  • The weight of an object acts through its center of gravity.

3. Conditions of Equilibrium:

  • Equilibrium: A rigid body is in equilibrium if it is at rest or moving with constant velocity.
  • Equations of equilibrium for a rigid body in two dimensions:
    • ΣFx = 0 (sum of forces in the x-direction is zero)
    • ΣFy = 0 (sum of forces in the y-direction is zero)
    • Στ = 0 (sum of torques about any point is zero)
  • Equilibrium of a rigid body under the action of three forces:
  • If three forces acting on a rigid body are concurrent (meet at a single point), then they are in equilibrium if they are co-planar and their vector sum is zero.
  • Lami’s theorem: If three coplanar forces acting on a rigid body are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

4. Applications of Equilibrium:

  • Levers:
  • A simple machine consisting of a rigid bar pivoted on a fixed point (fulcrum).
  • Used to multiply force or change the direction of a force.
  • Examples: Crowbar, seesaw, scissors
  • Pulleys:
  • A simple machine consisting of a wheel and a grooved rim.
  • Used to change the direction of a force or to multiply force.
  • Examples: Flagpole, elevator, clothesline
  • Inclined planes:
  • A sloping surface.
  • Used to reduce the effort required to lift an object.
  • Examples: Ramp, staircase, conveyor belt
  • Simple machines:
  • Devices that make work easier by changing the direction or magnitude of a force.
  • Examples: Lever, pulley, inclined plane, wedge, screw, wheel and axle

References:

  • NCERT Physics, Class 11, Chapter 10: Mechanical Properties of Solids
  • NCERT Physics, Class 12, Chapter 5: Laws of Motion
  • NCERT Physics, Class 12, Chapter 8: Gravitation