NLM

THEORY OF NEWTONS LAWS OF MOTION NLM-1 ACCELERATED FRAME OF REFERENCE  NEET 2020 JEE 2020

ACCELERATED FRAME OF REFERENCE

When Newton’s laws of motion were introduced in lesson 1, we emphasized that the laws are valid

only when observations are made in an inertial frame of reference (frame at rest or moving with

uniform velocity). Now we analyze how an observer in accelerated frame of reference

(Non-inertial frame) would attempt to apply Newton’s second law. 

    Once a frame of reference begins to accelerate the frame becomes non-inertial and Newton’s laws

do not hold good any more. To understand this in a better way, let us consider the rail-car. Suppose

a body is placed on the floor of the car which we consider as smooth. The train is moving with

uniform velocity and hence the position of the body with respect to the frame of reference

attached to the car remains constant. Suppose brakes are applied and the train begins to

decelerate. The body which was at rest on the floor, suddenly begins to slide along the

floor in the forward direction even though no force of any kind acts on it. Newton’s laws

seem to have been violated. Conventionally we would explain this motion as due to Newton’s

first law and the body due to the absence of friction continues to maintain its state of uniform

motion along a straight line with respect to the railway track. The train has now become a

non-inertial frame. 

    Non-inertial frames of reference are the system which are accelerated (or decelerated)

. Newton’s laws especially first and second cannot hold good for accelerating frames of

reference. Anyhow the Newton’s laws of motion can be made applicable to them by

applying an imaginary force on the body considered. This imaginary force is called inertial

force or pseudo-force or fictitious force. The magnitude of the force is the product of mass

of the body and the acceleration of the reference system. Its direction is opposite to the

acceleration of the reference. 

If a body of mass M is observed from a frame having acceleration then

            …(1)

    It should be emphasised again that no such force actually exists. But once it is introduced Newton’s

laws of motion will hold true in a non-inertial frame of reference. 

    Therefore for non-inertial frame, we can write 

        ,     …(2)

    where is acceleration of body with respect to frame. 

Question:    A pendulum is hanging from the ceiling of a car having an acceleration a0 = 10 m/s2 with respect to the road. Find the angle made by the string with vertical at equilibrium.(g = 10 m/s2)

Solution:    The situation is shown in figure. Suppose the mass of bob is m and the string makes an

angle θ with vertical, the forces on the bob in the car frame (non-inertial frame) are indicated. The forces are 

        (i) tension in the string 

        (ii) mg vertically downwards 

        (iii) ma0 in the direction opposite to the motion of car (pseudo force).

Writing the equation of equilibrium             T sin θ = ma0             T cos θ = mg         ∴  = 45°     ∴  the string is making an angle           45° with vertical at equilibrium

Question:    A block slides down from top of a smooth inclined plane of elevation θ=45° fixed in an elevator going up with an acceleration a0 = g. The base of incline has length L = 5m. Find the time taken by the block to reach the bottom.

Solution:     Let us solve the problem in the elevator frame. The free body force diagram is shown. The forces are 

        (i)    N normal to the plane

        (ii)    mg acting vertically down

        (iii)    ma0 (pseudo force).

            If a is the acceleration of the body with respect to incline, taking components of forces parallel to the incline

            mg sinθ + ma0 sinθ = ma

        ∴    a = (g + a0) sinθ

        This is the acceleration with respect to elevator.

The distance traveled is . If t is the time for reaching the bottom of incline                          = 1 s