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Equilibrium of Rigid Bodies

A body in which the distance between the constituent particles remain constant under the action of external forces is called a rigid body.

When a number of forces or torques act on a body and if the body is at rest or its center of mass moves with uniform speed the body is said to be in equilibrium.

If the resultant force and torque acting on a body is zero and if the body is at rest then it is said to be in static equilibrium.

Under the same conditions if the center of mass of the body moves with uniform speed it is said to be in dynamic equilibrium.

Example
In the case of a simple pendulum the forces acting on the bob are the tension T in the string acting vertically upwards and the weight W of the bob acting vertically downwards.

When the pendulum is at rest the bob is said to be in static equilibrium, because it is at rest under the action of the two forces.
Even when the pendulum is oscillating, T = W at the mean position A.

The bob overshoots the position A with a constant velocity and now the bob is said to be in dynamic equilibrium.

Whenever the resultant force and the resultant torque on a body is zero, the body is said to be in equilibrium.

When the body is in equilibrium, it can be either at rest or move (its center of mass) with a constant velocity (then linear acceleration is zero) or rotate with a constant angular velocity (then angular acceleration is zero).

Hence there can be a a translational equilibrium and a rotational equilibrium

Conditions for Translational Equilibrium of a Rigid body under Coplanar Forces

The first condition for a rigid body to remain in equilibrium is that the vector sum of all the forces acting on the body must be zero.

When only two or three forces act on an object, then the condition for equilibrium can be arrived at by considering the parallelogram law of force or law of triangle of forces.

But when a large number of forces are acting on a body the condition for equilibrium is arrived at by using the principle of resolution of forces, into rectangular components.

Consider a body acted upon by a number of coplanar forces say F1, F2, F3, F4 ………. Let the X-axis be parallel to the one of the forces say F1 and Y-axis be perpendicular to F1.

Other forces are resolved into two components, one component parallel to X-axis and the other component along the Y-axis. 

The algebraic sum of the forces acting along + X-axis and - X-axis are found out. Let it be Fx

Similarly the algebraic sum of the Y-component of the forces Fx is also obtained.


If there are only three coplanar forces Fx and Fy will contain only three terms.

Let Fx = Rx and Fy = Ry


The resultant of these components Rx and Ry is given as F =


Suppose the body has an acceleration ax along X-axis and ay along Y=axis.


Using Newton's second law, We have

F = ma

Fx = max, and Fy = may


When the body is in equilibrium ax = 0, ay = 0

Fx= 0, Fy = 0


Hence the condition for a rigid body under the action of a number of coplanar forces are

  1. Fx = 0
  2. Fy = 0




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