FORCE

We know by experience that all bodies in nature interact in some way with one another. Force is a measure of the interaction of bodies or of the particles of which the bodies consist. The force may produce either deformation (change in the size or shape of bodies) or acceleration (change in magnitude or direction of velocity).

Force is a vector quantity. Every force has a definite direction and the result of its action depends on the direction and the magnitude of the force.

If several forces are applied to a particle, they can be replaced by a resultant force. This resultant is the vector sum of the forces and can be found by any of the methods illustrated earlier.

SOME COMMON FORCES 

There are basically five forces which we commonly encounter in mechanics problems, namely Weight, Normal Force, Friction Force, Tension and Spring Force.

Weight
The weight W⃗  of a body is a force that pulls the body directly towards the earth. The force is due to gravitational attraction between two bodies. We will discuss this in detail later. Here, we consider an object of mass m located at a point where the free fall acceleration has magnitude g. Then the magnitude W of the weight is

W=mg

Its direction being vertically downward (towards the centre of the earth).
Normally we assume that weight is measured in an inertial frame. If it is measured in a non-inertial frame, it is called apparent weight.

Contact Forces
Whenever two surfaces are in contact they exert forces on each other. Such forces are known as contact forces. It is convenient to resolve these contact forces into components, one parallel to the contact surface, the other perpendicular to that surface.

Normal Force
The normal force is the component of the contact force that is perpendicular to the surface. It is a measure of how strongly the surfaces in contact  are pressed together. As an example push your hand straight down on the table. The force you feel resisting your push is the normal force of the table pushing up on your hand.

Frictional Force
The component of the contact force parallel to the contact surface is called frictional force. The direction of the frictional force is opposite to the relative motion (or attempted motion) of the two surfaces in contact.

Tension
The force exerted at any point in the rope/string/wire/rod is called the tension at that point. We may measure the tension at any point in the rope by cutting a suitable length from it and inserting a spring scale; the tension is the reading of the scale. The tension is same at all points in the rope only if the rope is unaccelerated and assumed to be massless.

Spring Force
As you may have discovered for yourself, springs resists attempts to change their length. In fact, the more you alter a spring’s length, the harder it resists. The force exerted by a spring may be represented as:

F=−kx

where x is the change in length, and k is the stiffness constant or simply, the spring constant. Unit of spring constant is N/m. This equation is also known as Hooke’s law. The minus sign in Hooke’s law shows that the direction of the force exerted by the spring is opposite to the displacement that produces it.
The spring constant depends on geometry of the spring and on the material property. For us, it is important to know that the spring constant is inversely proportional to its length, other things remaining the same. i.e.

k∝1l   (everything else constant)

Therefore if you cut a spring into two parts whose length are in ratio 1 : 2, their spring constants will be in ratio of 2 : 1. As in case of rope, we will usually deal with a massless spring, the force at each point of which is the same. Such spring and ropes are normally referred to as ideal.

NEWTON’S LAWS

Newton’s First Law

When there is no net force on an object
– an object at rest remains at rest, and
– an object in motion continues to move with a velocity that is constant in magnitude and direction.

Note
1) Newton’s first law really describes a reference frame. The property of the body to remain at rest or to retain its uniform linear motion in the absence of applied force is called inertia.
The first law is often called the law of inertia and the reference frames to which it applies are termed as inertial reference frames.
Thus an inertial reference frame is one which is either at rest or moves with a constant velocity relative to earth. Truly speaking, the earth itself is not an inertial reference frame (because it rotates as well as moves round the sun in an orbit) but for most practical purposes we can treat it as an inertial reference frame.

2) This law does not differentiate between objects at rest and objects moving with constant velocity. Indeed, an object moving with constant velocity in one inertial reference frame can be at rest in another inertial reference frame.

3) No net force here may mean the absence of all forces or the presence of forces whose resultant is zero.

Newton’s Second Law

We know from the first law, what happens when there is no unbalanced force on an object: its velocity remains constant. Now let us see What happens when there is an unbalance force on an object ? The Newton’s second Law gives answer to this question, that is, net force acting on a body will produce an acceleration.
When there is a constant unbalanced force on an object, the object moves with constant acceleration. Furthermore, if the force varies, the acceleration varies in direct proportion with larger force producing larger acceleration. Twice the force produces twice the acceleration in the same mass.

The magnitude of the acceleration produced depends on the quantity of matter being pushed. The quantity of matter is referred to as the inertial mass.
Newton’s second law states the relation between the net force and the inertial mass.

∑F⃗ =ma⃗ 

Note that the direction of acceleration is in the direction of the net force.
In terms of components
∑Fx=max      ∑Fy=may      ∑Fz=maz

Newton’s Third Law

Experiments show that forces occur in pairs. If you push against a wall, the wall pushes back at you. If one body A applies a force F⃗ BA on another body B, body B applies an equal but oppositely directed force F⃗ AB on  A i.e.

F⃗ BA=−F⃗ AB

Normally, one of these force (it does not matter which) is called the action force and the other is called the reaction force.  Thus the third law is also sometimes stated as “To every action there is always an equal and opposite reaction”. Note that the action and reaction always act on different objects.

FREE BODY DIAGRAM

In the application of any of the Newton’s laws, it is absolutely necessary to account correctly for all forces acting on the particle. The only forces that we may neglect are those whose magnitudes are negligible compared with other forces acting, such as the force of mutual attraction between two particles compared with their attraction to a body having large mass such as the earth. The only reliable way to account accurately and consistently for every force is to isolate the particle under consideration from all contacting and influencing bodies and replace the bodies removed by the forces they  exert on the particle isolated. The resulting free-body diagram (F.B.D.) is the  means by which every force, known and unknown, that acts on the particle is represented and hence account for. Only after this vital step has been completed should the appropriate equation or equations of motion be written.

As a part of the drawing of a free-body diagram, the coordinates axes and their positive directions should be clearly indicated. Also, the acceleration components are indicated by the side of the free-body diagram.

EQUILIBRIUM OF BODIES

A body or a system is said to be in equilibrium if it does not tend to undergo any further change of its own. Any further change must be produced by external means (e.g. force).
A body is said to be in translational equilibrium if the sum of the forces acting on the body is zero.

∑F⃗ net=0

If the particle is at rest, it is in static equilibrium. If it is moving at constant velocity, it is in dynamic equilibrium.  In either case ∑F⃗ =0. In terms of components ∑Fx=0, ∑Fy=0 and ∑Fz=0.
The simplest kind of equilibrium situation is one where two forces act on a body. When you stand motionless, you experience the downward gravitational pull of the earth, your weight W⃗ . The weight is balanced by an upward force exerted on you by the floor. This force is perpendicular to the floor and it is called the Normal Force N⃗ . Note that although N⃗  and W⃗  are equal and opposite, they do not constitute an action-reaction pair.Some other examples of static equilibrium are shown in the following figures.

Application 1
A block of mass 10 kg is suspended with two strings, as shown in the figure. Find the tensions T₁ and T₂  in the two strings.

Solution:

The free body diagram of the joint O is drawn as shown in the following figure.

Applying equations for equilibrium.
             ∑Fx=0T2sin30∘−T1=0                     ..(1)
∑Fy=0T2cos30∘−T1=0                  ..(2)
Thus,     T2=100cos300=2003√N
Substituting the value of T₂ in equation (1), we get
T1=T2sin30∘=1003√N

Application 2

Find the magnitude of the horizontal force F required to keep the block of mass m stationary on the smooth inclined plane as shown in the figure.

Solution:

The forces acting on the block are shown in the free body diagram.   Applying equations for equilibrium,
∑Fx=0Fcosθ−mgsinθ=0
or       F=mgsinθcosθ=mgtanθ