SATHEE: Equilibrium Of A Rigid Bodymoments And Center Of Gravity Topic

Equilibrium of a Rigid Body - Moments and Center of Gravity

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.

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.

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.

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

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