Center of Gravity, Center of Mass and Equilibrium | Physics Grade XI

Equilibrium

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Center of Gravity, Center of Mass and Equilibrium | Physics Grade XI

Center of Gravity, Center of Mass and Equilibrium

Center of Gravity
The center of gravity of a body is a point where the weight of the body acts and total gravitational torque on the body is zero.
Shape of the body and the position of its center of Gravity

S.N.

Shape of a body

Position of C.G.

1.

The uniform bar

Middle point of the bar

2.

Circular ring

Center of ring

3.

Circular disc

Center of disc

4.

Sphere, hollow sphere and annular disc

At center

5.

Cubical or rectangular block

Point of intersection of diagonals

6.

Triangular lamina

Point of intersection of medians

7.

Square lamia, rectangular and parallelogram lamina

Point of intersection of diagonals

8.

Cylinder

Middle point of axis

9.

Cone or pyramid

On the line joining the apex to the center of base at a distance equal to ¼ of the length of this line from the base

Center of mass
The center of mass of a body is the point which behaves as if all the mass of the body is concentrated at that point. 
 
Equilibrium of Concurrent Forces
A number of forces acting at a point are called concurrent forces. If the resultant of the concurrent force acting on a body is zero, the body is in equilibrium.
 
Equilibrium of rigid bodies
The rigid body must be in equilibrium if the following two conditions are met.
  1. The vector sum of the forces acting on the body must be zero
  2. The net torque acting on a body must be zero
Translational Equilibrium
For an object to be in a translational equilibrium the vector sum forces acting on it must be zero.
     i.e. net external force F = 0
Stating that the vector sum of forces acting on an object is zero is equivalent to;
     Fx = 0, Fy = 0 and Fz = 0
Where Fx, Fy and Fz are the component of a force in three perpendicular directions. This means that all the forces acting on one particular axis adds to zero. So, according to newtons second law of motion,
     F = ma
For translational motion. F = 0, so
     or, 0 = ma
     or, a = 0
     or, v = constant
Thus, when a body is in translational equilibrium, it is either at rest or it is moving with constant velocity.
 
Rotational Equilibrium
For a body to be in Rotational equilibrium, there may be three types of equilibrium.
  1. Stable equilibrium: A body is said to be in stable equilibrium if it returns to its equilibrium position after it has been displaced slightly. In this equilibrium, CG of the body lies at low point and when the body is displaced its CG lies at higher point than before.
  2. Unstable Equilibrium: A body is in unstable equilibrium if it does not return to its equilibrium state after it has been displaced slightly and it also doesn’t remain in displaced position.
  3. Neutral Equilibrium: A body is in neutral equilibrium if it always stays in the displaced position after it has been displaced slightly. The height of CG remains at same position in all the displaced positions.

Conditions for a body to be in Stable Equilibrium

  1. The C.G. of the body should lie as low as possible
  2. The base of the body should be as large as possible
  3. C.G should lie within the base of the body on displaced position
Lami’s Theorem
It states that, "if a particle is in equilibrium under the action of three concurrent forces, then each force is proportional to sine of the angle between the other two".

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