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Equilibrium of Concurrent Forces

A body in which the distance between the constituent particles remains 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 centre 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 centre 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 when the bob is at rest is nothing but the tension T in the string which acts 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 that balance each other.
 

When the pendulum is oscillating, T = W at the mean position A. The bob passes through 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 centre 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 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 are acting on a body, then the condition for equilibrium can be arrived using parallelogram law of forces or triangle law 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 us take X-axis parallel to one of the forces say F1 and Y-axis perpendicular to F1. Other forces are resolved into two components, one component parallel to X-axis and the other component along the Y-axis.

 

Depending on the direction of the forces the components may be parallel to + X-axis, or –X-axis and parallel to +Y-axis or –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 Fy is also obtained. If there are only three coplanar forces Fx = and Fy = Ry
 

The resultant of the 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 is

(i) Fx = 0 (ii) Fy = 0

 

Condition for Rotational Equilibrium of a rigid body


The resultant torque acting on the body must be zero. For this condition to be attained, the sum of anticlockwise torques about any axis must be equal to the sum of clockwise torques about the same axis. This means algebraic sum of all the torques acting on the body about the axis of rotation must be zero.

The general conditions of equilibrium are


i) The algebraic sum of the resolved components of the forces in any fixed direction must be zero.

ii) The algebraic sum of the resolved components of the forces in a perpendicular direction must be zero.

iii) The algebraic sum of the torques or moments of the forces about any point in their plane must be zero.


Let us consider a special case of equilibrium of a body acted upon on a number of coplanar parallel forces.


Let P, Q, R and S be four coplanar parallel forces acting on a body as shown in the figure. Using the condition number on, P+ Q- R-S = 0 i.e. the algebraic sum of the forces acting one the body is zero.


Using third condition, the anticlockwise moment = the clockwise moment.


Taking moments about the point A

P × AD + Q ×AF = R ×AC + S × AE

P ×AD + Q × AF – R × AC - S × AE = 0


Thus the algebraic sum of the moments of the forces acting on the body about any point is zero.

Common Forces in Mechanics


The origin and progress of the universe is very closely related to the forces between the particles in nature.


To study the physical interaction and to study about the motion of body knowledge of force is essential. This knowledge helps us to predict the future of the universe. There are four different types of interactions in the universe: gravitational, electromagnetic, weak and the strong interactions.
 

Forces, which are exerted only when two bodies are in contact, are called contact forces.


Examples: 

(i) The normal force exerted by a table on a book at rest on it

(ii) The force exerted by a compressed spring on a body attached to it

(iii) The pull of a string

(iv) The force of sliding friction

(v) A push or a pull

The contact force arises from the short range electrical and magnetic forces acting among the constituent atoms of the body. The tension in a string, the normal forces exerted by a surface on a body in contact with it, frictional force, and viscous force originate from intermolecular force. These forces are electromagnetic in nature.

The Gravitational Force


The force of attraction between any two material bodies in the universe called the gravitational force. This force of attraction between any two objects in the universe is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. This force acts on the line joining the centers of masses of the two bodies i.e.,

F N
F = G N

 

where m1 and m2 are two point masses separated by a distance r and G is the universal gravitational constant.

It is a weak force. Gravitational pull of the earth on moon provides the centripetal force for the moon to go around the earth. The birth and death of stars is governed by this force.

The Electrostatic Force


The force experienced by the two electric charges at rest is called the electrostatic force. Like charges repel, unlike charges attract. The force of attraction or repulsion between two charges is directly proportional to the product of charges and inversely proportional to the square of the distance between them. This is coulomb’s law.

i.e., F N
F = N

where q1 and q2 are two charges separated by distance r and ε 0 is a constant called the permittivity of free space.

The Magnetic Force


The force experienced by the two magnetic poles is called magnetic force. Like poles repel, unlike poles attract. The magnetic force is due to charges in motion.

The Electromagnetic Force


When a moving electric charge enters a magnetic field it experiences a force. The moving charge produces magnetic field. Thus the electric and magnetic forces together are called electromagnetic force.

Strong or Nuclear Forces


The force of attraction experienced by the two nucleons (neutron, proton) in the nucleus is called nuclear force. It is the strongest attractive force. It is a short range force. It does not depend upon the nature of charges.

Super force


Einstein tried to interpret the four basic forces as different components of a super force. But he could not. In 1970s physicists showed that the e. m. force and the weak force could be treated as different aspects of a single force called the electro weak force. 

Unification Theories


Theories like GUT’s (grand unification theories) try to unify the electro weak force and the strong force. The theories trying to unify all forces within a single frame are called super symmetry theories. According to string theorists, unification of all the forces are possible in a multi-dimensional space instead of a four dimensional space. Such theories are called superstring theories.





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