Inertial frame of references are those frames of reference in which newton's first and second law of motion is always hold true

A frame of reference in which Newton's law are not valid is called non-inertial frame of reference

In an inertial frame if a body is not acted by external force ,it continues to be in state of rest or uniform translatory motion.Thus in an inertial
frame if the body is not acted upon by an external force then acceleration would be zero a=(d^{2}r/dt^{2})=0

If a frame is inertial frame ,then all those frames which are moving with constant velocity relative to the previous frame are also inertial frames

Inertial frame of reference are necessary unaccelerated frames because if the frame is accelerated the particle moving with uniform velocity will appear

(9) Fictitious ( or Pseudo ) Forces

We already know about Non-inertial frame of reference .All the accelrated and rotatig frame of reference are non-inertial frame of refrence

Consider an interial frame of reference S and let S^{'} be any other frame moving with accleration w.r.t to frame S as shown in the below figure

Now if no external forces are acting on particle P .Then its acceleration would be zero in Frame S but in frame S^{'},an observer will find an acceleration -a_{0} acting on the particle.

The observer force on particle P of mass m in Frame S^{'} is -ma_{0}

But in reality no such force is acting on the particle and particle appears to be accelerated in this non-inertial frame of reference.Such one force is
known as Pseudo or Fictitious Force. Hence Pseudo Force on particle is F_{P}=-ma_{0}

Now if we apply F_{i} on the particle and a_{i} is the observed acceleration of particle in S frame(Inertial frame) The according to
Newton's law F_{i}=ma_{i}

For calculating net force in accelerated frame consider both the frames S and S^{'} coincide at time t=0 .After time t let r_{i} and
r_{n} be the position vector of the particle in frame S and S^{'} respectively
The relation between r_{i} and r_{n} is r_{i} =r_{n} +(1/2)a_{0}t^{2}
Where a_{0} is the acceleration of frame S^{'} wrt frame S
Differentiating the equation w.r.t time twice a_{i}=a_{n} + a_{0}
or ma_{i}-ma_{0}=ma_{n}
=> F_{0} + F_{P}=F_{N}
This equation gives observed force in accelerated frame of reference