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