Force analysis in accelerated frame  (Page 2/3)

Resolution of differences

The two observes meet and exchange their observations and discuss the differences. It is obvious that the observer on the ground (inertial frame of reference) prevails as his analysis of force yields the correct value of the normal force in the accelerated lift. Clearly, observer in the accelerated frame of reference (no-inertial) needs to do some adjustment, while applying Newton's second law of motion. The observer of inertial frame of reference points out to the observer in the accelerated frame of reference that the lift was actually being pulled up by an external force and that this external force needs to be accounted in the analysis by the observer in the lift.

They conclude as :

1: Newton’s second law is valid in inertial frame of reference and needs to be modified, when used in the accelerated frame of reference.

2: The body need to be subjected to an additional force, usually referred as pseudo force, to account for the acceleration of the frame of reference. Once this additional force is applied on the body, Newton's second law can be applied as usual even in the non-inertial frame of reference.

3: The application of pseudo force should be carried out in the following manner :

• Determine magnitude of pseudo force," $F_S$ ", by multiplying the mass of the body "m" with the acceleration “a” of the accelerated frame of reference,
• Apply pseudo force on the body in the direction opposite to that of the acceleration “ a ”.
• Analyze the situation, using Newton’s second law as if the frame of reference was an inertial frame of reference.

In the nutshell, the pseudo force is given by following vector relation,

where " a " is the acceleration of non-inertial frame of reference.

The two observers also conclude that the spring balance does not measure weight of the block (mg), but the normal force applied on it. Besides they realize that normal contact force depends on the acceleration of the moving reference (container).

Let us check the analysis of the observer in the lift, using the concept of pseudo force. Now, the free body diagram of the block with pseudo force is as given here (the downward forces are shown as two parallel vectors only to show them individually).

$$\begin{array}{l}\sum F_y=N-mg-ma=0\\ \Rightarrow N=m(g+a)=10(10+2)=120\mathrm{Newton}\end{array}$$

Note that while applying law of motion in the accelerated frame, the observer has not modified his observation that the block is at rest. Adoption of pseudo force technique simply allows reconciling force analysis with the observed value measured in the accelerated system, using Newton’s second law.

Theoretical basis of analysis in accelerated frame

The conception of pseudo force in an accelerated frame of reference is not without basis. We can develop the theoretical framework that justifies using additional force in accelerated frame for applying Newton’s second law.

Here, we consider a body “A” of mass “m”. An external force “ F ” acts on the body as shown. The acceleration of the body with respect to ground, “ $a_A$ ”, is related to external force as :