Physics Motion in straight line

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Motion in straight line

Frame of Reference

A system of co-ordinates whose axes can be suitably chosen is said to be a frame of reference. For location of a point -P- we need three co-ordinate x, y and z. For complete identification of an event we must know -t- also, i.e., the time of the occurrence. Hence an event in characterized by four co-ordinates (x,y,z,t). A reference frame describing an event in these four co-ordinates is known a space time frame.

A system of co-ordinates whose axes can be suitably chosen is said to be a frame of reference. For location of a point -P- we need three co-ordinate x, y and z. For complete identification of an event we must know -t- also, i.e., the time of the occurrence. Hence an event in characterized by four co-ordinates (x,y,z,t). A reference frame describing an event in these four co-ordinates is known a space time frame.

Inertial and Non-Inertial Frame of Reference

(a) Inertial Frame:-

A frame of reference either at rest or moving with a uniform velocity (zero acceleration) is known as inertial frame. All the laws of physics hold good in such a frame.

(b) Non-Inertial or Accelerated Frame:-

It is a frame of reference which is either having a uniform linear acceleration or is being rotated with uniform speed.

(a) Inertial Frame:-

A frame of reference either at rest or moving with a uniform velocity (zero acceleration) is known as inertial frame. All the laws of physics hold good in such a frame.

(b) Non-Inertial or Accelerated Frame:-

It is a frame of reference which is either having a uniform linear acceleration or is being rotated with uniform speed.

Simulation for Frame of Reference

(i) Reference frame is at rest:-
The acceleration of the mass will be, say, `vec{a}_{rest}.`

Therefore the force on it will be `vec{F}_{rest}.`

We will reason that

`vec{F}_{rest} = m\vec{a}_{rest}`

(ii) Reference frame starts moving with constant velocity vector-v :-

The acceleration of frame `= vec{a} = 0`

Thus, acceleration of mass m relative to frame is given by

`vec{a}_{i n ertial} = vec{a}_{rest} - vec{a} = vec{a}_{rest}`

Force on it will be `vec{F}` inertial and we will reason that

`vec{F}_{i n ertial} = mvec{a}_{i n ertial} =m vec{a}_{rest} = vec{F}_{rest}`

(iii) Reference frame moves with constant acceleration:-

Let the acceleration of frame be `vec{a}_{f rame} .`

Thus, acceleration of mass relative to frame will be `vec{a}_{rel}.`

`vec{a}_{rel} = vec{a}_{i n ertial} - vec{a}_{f rame} = vec{a}_{rest} - mvec{a}_{f rame}`

Let there be force `vec{F}` frame on mass we will reason, that

`vec{F}_{frame} = m vec{a}_{rel} = m vec{a}_{rest} - m vec{a}_{f rame}`

`= vec{F}_{rest} + m(-vec{a}_{f rame}) = vec{f}_{rest}+vec{F}_{pseudo}`

We see that the force is not the same as that in the inertial frames.

Therefore we postulate that under observation from an accelerated reference frame we substitute the inertial forces on the body with those same initial forces plus an additional force which numerically equal to the mass of the body under observation times the acceleration of the frame taken in the opposite direction. This force we call as pseudo force.

(i) Reference frame is at rest:-
The acceleration of the mass will be, say, `vec{a}_{rest}.`

Therefore the force on it will be `vec{F}_{rest}.`

We will reason that

`vec{F}_{rest} = m\vec{a}_{rest}`

(ii) Reference frame starts moving with constant velocity vector-v :-

The acceleration of frame `= vec{a} = 0`

Thus, acceleration of mass m relative to frame is given by

`vec{a}_{i n ertial} = vec{a}_{rest} - vec{a} = vec{a}_{rest}`

Force on it will be `vec{F}` inertial and we will reason that

`vec{F}_{i n ertial} = mvec{a}_{i n ertial} =m vec{a}_{rest} = vec{F}_{rest}`

(iii) Reference frame moves with constant acceleration:-

Let the acceleration of frame be `vec{a}_{f rame} .`

Thus, acceleration of mass relative to frame will be `vec{a}_{rel}.`

`vec{a}_{rel} = vec{a}_{i n ertial} - vec{a}_{f rame} = vec{a}_{rest} - mvec{a}_{f rame}`

Let there be force `vec{F}` frame on mass we will reason, that

`vec{F}_{frame} = m vec{a}_{rel} = m vec{a}_{rest} - m vec{a}_{f rame}`

`= vec{F}_{rest} + m(-vec{a}_{f rame}) = vec{f}_{rest}+vec{F}_{pseudo}`

We see that the force is not the same as that in the inertial frames.

Therefore we postulate that under observation from an accelerated reference frame we substitute the inertial forces on the body with those same initial forces plus an additional force which numerically equal to the mass of the body under observation times the acceleration of the frame taken in the opposite direction. This force we call as pseudo force.

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