Force analysis in accelerated frame

$$F=m{a}_A$$

We, now, seek to analyze the motion with respect to a frame of reference, which is moving with acceleration, ” $a_B$ ” with respect to ground reference. Let the acceleration of the body with respect to this accelerated frame be “ $a_{\mathrm{AB}}$ ” .

The product of the mass and the acceleration of the body with respect to accelerated reference is given by :

$$F\prime =m{a}_{AB}$$

This product is not equal to the product as obtained in inertial ground frame. This is because, acceleration of the body with respect to accelerated reference “ $a_{\mathrm{AB}}$ ” is given as :

$$a_{AB}=a_A-a_B$$

As such, the product in the accelerated frame evaluates to :

$$\Rightarrow F\prime =m{a}_{AB}=m{a}_A-m{a}_B=F-m{a}_B$$

If Newton’s second law of motion is valid in the accelerated frame, then it should connect external force on the body with the acceleration, “ $a_{\mathrm{AB}}$ ”, in the accelerated frame of reference. Clearly, this is not the case. However, if we replace external force “ F ” by “ $F-m{a}_B$ ”, then the modified external force is equal to the product of mass and acceleration of the body in the accelerated frame.

Clearly, we need to apply a force " $a{a}_B$ "maB in the direction opposite to the acceleration of the frame of reference “B”. This force is known or termed as “pseudo” force.

Problem 1 : A pendulum is suspended from the roof of a train compartment, which is moving with a constant acceleration “a”. Find the deflection of the pendulum bob from the vertical as observed from the ground and the compartment.

Solution : We analyze the problem from the perspectives of both inertial and accelerated frames.

(i) In the inertial frame of reference,

$\text{Free body diagram of pendulum bob}$

The bob has acceleration “a” towards right. The forces on the bob are (i) weight of the bob, mg, and (ii) tension in the string.

$$\begin{array}{l}\sum F_x=T\sin \theta =m{a}_x\\ \Rightarrow T\sin \theta =ma\end{array}$$

and

$$\begin{array}{l}\sum F_y=T\cos \theta -mg=0\\ \Rightarrow T\cos \theta =mg\end{array}$$

Combining two equations, we have :

$$\begin{array}{l}\tan \theta =\frac{a}{g}\\ \Rightarrow \theta ={\tan }^{-1}\left(\frac{a}{g}\right)\end{array}$$

(ii) In the accelerated frame of reference,

$\text{Free body diagram of pendulum bob}$

The bob is at rest. The forces on the bob are (i) weight of the bob, mg, (ii) tension in the string and (iii) pseudo force ma.

$$\begin{array}{l}\sum F_x=T\sin \theta -ma=0\\ \Rightarrow T\sin \theta =ma\end{array}$$

and

$$\begin{array}{l}\sum F_y=T\cos \theta -mg=0\\ \Rightarrow T\cos \theta =mg\end{array}$$

Combining two equations, we have :

$$\begin{array}{l}\tan \theta =\frac{a}{g}\\ \Rightarrow \theta ={\tan }^{-1}\left(\frac{a}{g}\right)\end{array}$$

Alternatives

Application of force analysis in accelerated frame of reference may have two approaches. We can analyze using Newton’s law from an inertial frame of reference. Alternatively, we can use the technique of pseudo force and apply Newton’s law right in the accelerated frame of reference as described above.

There is a school of thought that simply denies merit in pseudo force technique. The argument is that pseudo force technique is arbitrary without any fundamental basis. Further, this is like a short cut that conceals the true interaction of forces with body under examination.

On the other hand, there are complicated situation where inertial frame approach may turn out to be difficult to work with. Consider the illustration depicted in the figure. Here, a wedge is placed on the smooth surface of an accelerated lift. We have to study the motion of the block on the smooth incline surface of the wedge.

Multiple accelerations here complicates the situation. The lift is accelerated with respect to ground; wedge is accelerated with respect to lift (as the surface of the lift is smooth) ; and finally block is accelerated with respect to wedge (as wedge surface is also smooth). In this case, it would be difficult to assess or determine attributes of motion by analyzing force in the inertial ground reference. In situation like this, analysis of forces in the non-inertial frame of reference of the lift eliminates one of the accelerations involved.

It must again be emphasized that when we analyze motion with respect inertial frame of reference, then all measurements are done with respect to inertial frame. On the other hand, if we analyze with respect to accelerated frame of reference using concept of pseudo force, then all measurements are done with respect to accelerated frame of reference. For example, if the analysis of force in non-inertial frame yields an acceleration of the block as 5 $m/s^2$ , then we must know that this is the acceleration of the block with respect to in incline i.e. accelerated frame of reference - not with respect to ground.

It is generally recommended that we should stick to force analysis in the inertial frame of reference, unless situation warrants otherwise.