EQUILIBRIUM OF FORCE

EQUILIBRIUM OF FORCES

An object is in equilibrium when it is not accelerated, that is there is no force acting on it in any direction. For a body in equilibrium, the forces acting on it are so related in magnitude and direction that no acceleration results.

A body is said to be in equilibrium when the body as a whole either remains at rest or moves in a straight line with constant speed and the body is either not rotating at all or is rotating at a constant angular velocity.

FORMS OF EQUILIBRIUM    

(a) Static equilibrium: This is when a body is at rest or moving with constant velocity.

(b) Dynamic or Kinetic equilibrium: This is when a body is moving with a constant velocity in a straight line or rotating with a constant angular velocity about a fixed axis through its centre of mass. This means that the forces that set the body in motion balance the forces that resist the motion.

RESULTANT AND EQUILIBRANT FORCES

A body acted upon by two or more forces is said to be in equilibrium if it does not move or rotate. Under this condition, the sum of the forces moving the body in one direction is equal to the sum of the forces pushing the body in the opposite direction.

In the force board experiment, the resultant R of two forces P and Q acting at point O was found by the parallelogram method to be equal in magnitude but exactly opposite in direction to the third force that kept the point O in equilibrium. This third force is known as EQUILIBRANT FORCE.

The Resultant force is that single force which acting alone will have the same effect in magnitude and direction as two or more forces acting together.

The Equilibrant of two or more forces is that single force which will balance all the other forces taken together. It is equal in magnitude but opposite in direction to the resultant force.

Example

Two forces 10 N each are inclined at 120⁰ to each other. Find the single force that will:

(i)                 Replace the given force system.

(ii)               Balance the given force system.

Solution

(i)                 The resultant force is given by:

R² = 10² + 10² + 10 x 10 Cos 120⁰  

     = 200 – 200 Cos 60⁰

     = 10 N.

(ii)               The equilibrant force is the force that will balance the given system of forces.

Equilibrant is equal and opposite of the resultant. Hence,

Equilibrant = 10 N directed opposite to the resultant R.

EQUILIBRIUM OF THREE FORCES ACTIONG AT APOINT

The three forces that keep a body in equilibrium can be represented in magnitude and direction by the three sides of a triangle as shown below

OB represents the force Q in magnitude and direction, OA represents the force P both in magnitude and direction and CO represents the equilibrium E in magnitude and direction. The triangle shown below shows below is the vector representation of the three forces acting at O.

The principle of triangle of forces states that if three forces are in equilibrium, they can be represented in both magnitude and direction by the three sides of a triangle taken in order.

MOMENT OF A FORCE

When we turn on a tap, tighten a nut with a spanner or screw a nail in or out of a wood with screw driver, we are exerting a turning force and producing a turning effect about a point along an axis. Such turning effect brought about in each case is called MOMENT OF A FORCE.

Two factors are involved in each case:

(i)                 The magnitude of force applied; and

(ii)               The perpendicular distance of its line of action from the axis or pivot about which the turning effect is felt or exerted.

Therefore, the moment of a force about a point or axis is defined as the product of the force and the perpendicular distance of its line of action from the point.

Moment = force x perpendicular distance of the pivot to the line of action of the force. It is vector quantity and its S.I. unit is Newton – meter (NM).

PRINCIPLE OF MOMENTS

The principle of moments states that if a body is in equilibrium then the sum of the clockwise turning moments acting upon it about any point equals the sum of the anticlockwise turning moments about the same point.

From fig. a above, clockwise moment about O is given as F₂x₂ while anticlockwise moment is F₁x₁. Therefore, the sum of moment is given as F₂x₂ - F₁x₁. If the bar is in equilibrium,

F₂x₂ - F₁x₁ = 0     

F₂x₂ = F₁x₁.

 In fig.b, taking moment about O,

F₁x₁ – F₂x₂ + F₃x₃ – F₄x₄.

 If the bar is in equilibrium,

F₁x₁ – F₂x₂ + F₃x₃ – F₄x₄ = 0

F₁x₁ + F₃x₃ = F₂x₂ + F₄x₄.

CENTER OF GRAVITY

The centre of gravity of a body is defined as the point through which its resultant weight acts.

If a body is supported at its centre of gravity, it will remain stable. The centre of gravity of any uniform symmetrical object can easily be determined, because it is at centre of the body, for example, the centre of gravity of a uniform meter rule is at 50 cm, circle is at the centre.

The centre of gravity of an irregular body can be found either by using plumb line or balancing method.

CONDITIONS FOR EQUILIBRIUM OF NON – PARALLEL FORCES

(a)  For the action of parallel coplanar forces

(i)                 The algebraic sum of the forces acting on the body in any direction must be zero. That is, Sum of upward forces = Sum of downward forces.

(ii)               The algebraic sum of the moments acting on the body in any direction must be zero. That is, Sum of CW moment = Sum of ACW moment.

(b) For the action of non – parallel coplanar forces

(i)                 The vector sum of all the forces acting on the body must be zero. That is, Σx = 0 and Σy = 0.

(ii)               The algebraic sum of the moments acting on the body in any direction must be zero. That is, Sum of CW moment = Sum of ACW moment.