# CLASSICAL MECHANICS | Virtual displacement| D’Alembert’s principle| Frame of Reference

## Virtual displacement

The virtual displacement is called ‘virtual” in the sense that there is no passage of time during such a displacement (But actual displacement dr does occur in a finite though small interval of time dt).

Virtual displacement is consistent with the forces and constraints imposed on the system at a given instant t.

Principle of virtual work — In any virtual displacement, the total a work done by forces of constraint is zero .

## D’Alembert’s principle

Newton’s second law of motion and principle of virtual work for system of particles are applicable to the statics which was formed by D’Alembert and known as D’Alembert’s principle.

In a system of particles total force Fi on the th particle may be split up into two parts ;force of constraint. ; applied force . ———————————– 1

According to Newton’s second law of motion, ————————————2

Thus work done by the virtual displacement (del)ri , of the system Virtual work done by forces of constraint being zero. D’Alembert’s principle.

To maintain dynamical equilibrium of a particle virtual work done by all applied force equal to reverse effective force provided by work done by the constraints vanishes.

#### Frame of Reference

Frame of reference : – The surrounding of a moving object relative to which its motion can be described is known as frame of reference.

So, Frame of reference are of two types

i) Inertial frame of reference

ii) Non inertial frame of reference.

##### i) Inertial frame of reference

A reference frame e.g. a coordinate system in which Newton’s first laws of motion holds good, is known as an inertial frame of reference

In an inertial frame a body continues in its state of rest or of uniform motion in a straight line as long as external force act on it.

N.B.– All the Newton’s law of motion hold good in an inertial frame, hence it is also called Newtonian frame of reference.

Example:

All the rest frames or all the frames of reference moving with a constant velocity with respect to an inertial frame are inertial frame. The best approximation of an inertial frame is the frame reference in the intergalactic space.

##### ii) Non inertial frame of reference

A reference frame e.g. a coordinate system in which Newton’s first law of motion does not hold good, is known non inertial frame of reference. In a non inertial frame a body is accelerated through an inertial frame is called non inertial frame of reference.

N.B.– Non-inertial frame also called Non Newtonian frame of reference

Examples :

A frame having uniform linear acceleration or a frame which is uniformly rotating is non inertial of reference.

##### Why earth is a non-inertial frame ?

To get this, first we have to imagine the rotation of earth.The earth is a rotating sphere as it is rotating about its own axis. This has a centripetal acceleration for rotation. that implies earth is a non inertial frame.

#### Show that frame of reference having a uniform translatory motion relative to an inertial frame is also inertial.

If an inertial frame S’ is moving relative to another inertial frame S with a uniform velocity v along a straight line,

then a = a’ i.e. the acceleration of a particle is same in the two inertial frames.

In other words, if a particle is unaccelerated in s frame, it is also un accelerated in S’ frame i.e. a particle at rest on in uniform motion in a straight line is S frame will also be at rest or in uniform motion in a straight line in S’ frame.

### What is fictitious force or pseudo force?

Consider a reference frame S’ moving with an acceleration ao relative to a frame S. If s is inertial then S’ in non-inertial frame.

Now consider a particle of mass m on which no external force is acting in the frame S.  Then the acceleration of the particle in S frame is zero but the particle will appear to have an acceleration ao relative to S’ frame.

Hence, for an observer in S’ frame, force acting on the particle Where as for   an observer in S frame,  force acting on on the particle is zero.

Such a force which appears only due to the acceleration of the frame of reference is called fictitious force or pseudo force.

Examples of fictitious force ;

i) Centrifugal force.

ii)Coriolis force.

#### Can particles be in equilibrium in a non inertial frame?

A particle is in equilibrium when no net force is acting on it. This condition is satisfied when the external force acting on the particle in a non inertial frame is equal and opposite to the fictitious force .

Total force acting on the particle in an non-inertial frame Where Fi – external force acting on the particle  as observed in an inertial frame.

F0- fictitious force as observed due to non-inertial frame.

When then i.e. the particle is in equilibrium. 