### Equilibrium //State of Equilibrium //Mechanics //Equilibrium of Concurrent Forces //Equilibrium of Rigid Bodies //Translational Equilibrium //Rotational Equilibrium //States of Equilibrium //Condition for a Body in Stable Equilibrium

Equilibrium of Concurrent Forces

A number of forces acting at a point are called concurrent forces. If the resultant of the concurrent forces acting on a body is zero, the body is in equilibrium. Figure : Figure showing equilibrium

When concurrent forces are in equilibrium, then the state of the body will not change i.e. if the body is at rest it remains at rest and if in uniform motion, continue to move with uniform velocity on straight line.

Equilibrium of Rigid Bodies

A rigid body will be in equilibrium if the following two conditions are met.

a).The vector sum of the forces acting on the body must be zero.

b).The net torque acting on the body must be zero.

The first condition ensures that the linear or translational acceleration is zero i.e. either the body is at rest or it is moving with uniform linear velocity. Under this condition, the body is said to be in translational equilibrium. The second condition ensures that the angular acceleration is zero i.e. either the body is at rest or it is moving with uniform angular velocity. Under this condition, the body is said to be in rotational equilibrium. A body satisfying both these conditions is said to be in equilibrium. If such a body is stationary, it is said to be in static equilibrium and if it is moving with uniform linear or angular velocity, it is said to be in dynamic equilibrium,

Translational Equilibrium

For an object to be in translational equilibrium the vector sum of the forces acting on it must be zero. i.e. net external force, F = 0.

Now, for x- axis

F(x). = 0,

Stating that the vector sum of forces acting on an object is zero is equivalent to

F(y) = 0, F(z), = 0. where F(x), F(y), and F(z), are the components of a force in three perpendicular directions. It means that all the forces along the x-axis add to zero. Same is true for forces along y and z-axis. From Newton's send 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 an object to be in rotational equilibrium, the net torque acting on it must be zero.

Net torque, T = 0

Stating that the net torque acting on an object is zero means that angular acceleration of the body about any axis of rotation is zero. So, the body is either at rest or moving with constant angular velocity about the axis.

States of Equilibrium

Bodies may be in three types of equilibrium. These are discussed below.

(a) Stable equilibrium: A body is in stable equilibrium if it returns to its equilibrium position after it has been displaced slightly as in figure below.

In this equilibrium, C.G. of the body lies low and when the body is displaced from its equilibrium position, the C.G. at this position is higher than before.

(b) Unstable equilibrium: A body is in unstable equilibrium if it does not return to its equilibrium position after it has been displaced slightly and does not remain in the displaced position as in figure given.

When the body is displaced form its equilibrium position, its C.G. is lower than before.

(c) Neutral equilibrium: A body is in neutral equilibrium if it always stays in the displaced position after it has been displaced slightly as in figure below. In this equilibrium, height of the C.G. of the body does not change but remains at the same height from the base in all displaced positions.

Conditions for a Body in Stable Equilibrium

Following conditions for a body to be in stable equilibrium.

(i) The C.G. of the body should lie as low as possible.

The body is in stable equilibrium when its C.G. lies as low as possible from the base. Due to this reason, the bottom of the ship is made heavy and the cargo is always kept at its base. This makes the ship more stable. A truck loaded with iron rod is more stable than a truck loaded with cotton or straw. It is due to the fact that when the truck is loaded with iron rod, its C.G. is lowered while when it is loaded with cotton, its C.G. rises and there is likely of toppling the loaded truck.

(ii) The base of the body should be as large as possible.

The body is in stable equilibrium when its base is large as possible. An animal with four legs is more stable as compared to a man because the base of the animal is large. Similarly, four wheeler vehicles are more stable than three wheeler and two wheeler vehicles because four wheeler vehicles have large base.

(iii) C.G. should lie with in the base of the body on displaced position.

A man carrying a bucket of water in his right hand leans towards the left hand side. Doing this, the vertical line through the C.G. will pass through the base. Due to the same reason, a man carrying a load on his back has to bend forward. If he wants to carry the load straight upon his back, the vertical line through the C.G. falls outside the base and cannot carry the load easily.

The vertical line through the C.G. of the leaning tower of Pisa passes within the base of the tower and so, the tower does not fall down.