# Statics of Rigid Bodies

A body is said to have attained mechanical equilibrium if it either remains at rest or moves with a constant velocity. The branch of physics which studies the conditions of equilibrium of a body at rest is called statics. Strictly speaking, all bodies which attain equilibrium under a set of forces are deformed to a certain extent. However, for comparatively small forces, we may ignore the small deformations and treat the body under investigation as a perfectly rigid body.

Let us consider a three-dimensional rigid body of arbitrary shape in a frame of reference, Oxyz. A set of n external forces F1, F2, …, Fn are acting on the body. The static equilibrium of the rigid body requires the following conditions to be satisfied separately. The condition of translational equilibrium is:

``` ∑ F = F1 + F2 + … + Fn = 0 ```

If τ1, τ2, …, τn are the individual torques produced by the above forces about the origin O, the condition of rotational equilibrium is:

``` ∑ τ = τ1 + τ2 + … + τn = 0 ```

The torque equation is valid for any choice of origin inside or outside the boundaries of the body, provided the force equation holds good.

The number of velocity components required to completely describe the motion of a body is called the number of degrees of freedom of the body. An insect has one degree of freedom while walking along a straight wire, two degrees of freedom while crawling all over a flat table top, and a maximum of three degrees of freedom while flying randomly in the air. Treating the insect as a particle, all these degrees of freedom are associated with translational motion only. An extended body on the other hand is capable of both translation and rotation. Each type of motion contributes one to three degrees of freedom. In the most general case, an extended body acted upon by a set of non-coplanar forces has six degrees of freedom – three translational and three rotational. Now, each degree of freedom corresponds to one independent condition of equilibrium. In other words, the number of degrees of freedom is equal to the number of conditions of static equilibrium of a rigid body. Our video lectures mostly deal with simple cases involving coplanar forces and few equations, such as the equilibrium of a leaning ladder.

A statically indeterminate problem is one in which the number of unknown quantities is higher than the number of independent equations of equilibrium involving those unknowns. Obviously, such a problem cannot be solved. This type of situation arises when a body gets one or more extra support(s) over and above the minimum required for its equilibrium. An interesting example is a ladder resting on a rough floor against a rough wall.

The centre of gravity of a body is a fixed point through which the resultant of the forces of gravity acting on the constituent particles of the body passes, regardless of how the body is oriented in space. The centre of gravity and the centre of mass of a body are located at the same point, provided the body is placed in a uniform gravitational field. For exceptionally large bodies, such as a mountain, these two points do not coincide.

The stability of static equilibrium of a body can be determined as follows. Let the body be slightly tilted from its position of equilibrium. If a restoring torque comes into play and brings the body back to its initial position, the equilibrium is stable. If a destabilising torque comes into play and pushes the body further from its initial position, the equilibrium is unstable.

#### Basic level videos

Conditions Of Static Equilibrium Of A Rigid Body 44:30 Basic
250 4
Various Cases Of Static Equilibrium In Two Dimensions I 1:20:40 Basic
350 7.5
Various Cases Of Static Equilibrium In Two Dimensions II 1:22:55 Basic
350 7.5
Equilibrium Of A Leaning Ladder 1:08:23 Basic
350 7.5
More Problems On Equilibrium Of A Leaning Ladder 1:06:36 Basic
350 7.5
Centre Of Gravity Of A Rigid Body 1:14:42 Basic
350 7.5
Stability Of Static Equilibrium 1:02:10 Basic
350 7.5

#### Advanced level Videos Note: (CE) Stands for Problems from Competitive Examination Papers

Advanced Level Problems On Statics Of Rigid Bodies I 01:21:41
350
7.5
Advanced Level Problems On Statics Of Rigid Bodies II 01:04:24
350
7.5
Advanced Level Problems On Statics Of Rigid Bodies III 01:06:37
350
7.5
Advanced Level Problems On Statics Of Rigid Bodies IV 55:41
300
5