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The following passage is extracted from the book "Concepts of Physics, Part 1" by H C Verma:

Let us consider the situation shown in figure. An elevator cabin falls down after the cable breaks. The cabin and all the bodies fixed in the cabin are accelerated with respect to the earth and acceleration is about 9.8 $m/s^2$ in the downward direction. Consider the lamp in the cabin. The forces on the lamp are
the gravitational force W by earth and the electromagnetic force T (tension) of the rope. The direction of W is downward and the direction of T is upward. The sum is (W-T) downward. Measure the acceleration of the lamp from the frame of reference of the cabin. The lamp is at rest. The acceleration of the lamp is zero. The person A who measured this acceleration is a learned one and uses Newton's first law to conclude that the sum of the forces acting on the particle is zero, i.e, W-T=0 or, W=T.

Instead, if we measure the acceleration from the ground, the lamp has an acceleration of 9.8 $m/s^2$. Thus "a" is not equal to zero and hence the person B who measured this acceleration, concludes from Newton's first law that the sum of the forces is not zero. Thus, W-T is not zero or W is not equal to T. If A measures acceleration and applies the first law he gets W=T. If B measures the acceleration and applies the same first law, he gets W to be not equal to T....
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The forces acting on the lamp as seen by B are tension (T) due to string, and gravitational force (W), which gets cancelled. I think there is another force (or acceleration) acting on the lamp (and all other bodies in the cabin), i.e due to acceleration of the cabin itself. So, thread and lamp gets accelerated at the same rate and the tension cancels with W. There seems to be no problem here.

But if we consider what textbook says, i.e with no force due to cabin motion itself and W to be not equal to T, as there are no other force acting, lamp should fall down. But this is not the case, the cabin's thread breaks but not of lamp's.

I am failing to see the mistake, are the textbook statements (in relation with my argument) correct?

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i think this has to do something with inertial and non-inertial frames, but am not really sure – Gowtham Jan 19 at 12:40
Also related fictitious forces – Gowtham Jan 19 at 13:02
Gotta be a multi-duplicate. – Carl Witthoft Jan 19 at 13:05
In the scenario described, from any frame of reference, there is no tension in the rope. – DJohnM Jan 19 at 14:34
That author is incredibly inept. – Hot Licks Jan 19 at 21:28

6 Answers 6

I'm quite sure that there's something wrong with the sentence "The person $A$ who measured this acceleration is a learned one and uses Newton's first law to..." in that book. The usage of Newton's first law (in fact all three) is restricted; it can only be used in an inertial reference frame. If you must,an inertial force shoud be added. If the reference you choose has an acceleration of $a$, then an inertial force $F = -m.a$ ('$-$'means the force is on the opposite direction of $a$,'$m$' being the mass of the object) should be acting on each object. For the case in your book,inertia force is $F = -mg = -W$ downward,so $$W - T - W = 0 \implies T = 0$$. I think your book is alerting readers by showing a 'classical mistake'.

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Text specifies a constant gravitational field. So Einsteins equivalence principle allows us to consider it as an inertial reference frame. – Taemyr Jan 19 at 14:24
well, "Einsteins equivalence principle".It says we can't tell the difference between acceleration and gravitation if we're unaware of outside situation.So we can consider an accelerating cabin as a gravitational field that equals to the acceleration but on the opposite direction.As we can see,using "Einsteins equivalence principle",there are two equal gravitational fields upward an downward from reference frame of the cabin,consequently the lamp is still "floating".So you see we use another theory and come to the same conclusion. – Noctihaar Surralyn Jan 19 at 14:59
I was reacting to your claim that the cabin is not an inertial frame. – Taemyr Jan 19 at 15:02
Pay attention to this:"It says we can't tell the difference between acceleration and gravitation if we're unaware of outside situation." Einsteins equivalence principle means acceleration = gravitational field,not acceleration = inertial frame.(ps:I'm not a native English speaker so it may take some time for me to express my idea.Hope you will undertand.) – Noctihaar Surralyn Jan 19 at 15:09
I am not convinced the distinction you specify exists. An inertial frame is one in which Newtons first law holds, ie. that objects with no net forces have 0 acceleration. Einsteins equivalence principle tells us that an freefalling reference through a flat gravitational field is equivalent 0 acceleration. – Taemyr Jan 19 at 15:17

From observer B's perspective the lamp is falling down. It's acceleration is 9.8 m/s. From B's perspective the lamp is not getting closer to the floor of the cabin, because the cabin is also accelerating at 9.8 m/s.

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Observer $A$ & Lamp have the same acceleration; they are in a non-inertial reference frame.

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Both observers will calculate the tension in the thread to be zero. Observer A inside the lift will measure W to be zero (as there appears to be no gravitational acceleration) and observe the lamp to be stationary, hence calculate W = T = 0. Observer B will observe the lift accelerating under gravity, observe the lamp accelerating at the same rate (hence remaining stationary with respect to the lift), and calculate that W - T = W hence T = 0.

This makes intuitive sense if you imagine replacing the thread with a spring: when the lift cable is cut and the lift starts to fall, the tension in the spring will pull the lamp up and cause it to bounce around. The only way this can not happen (and so for the lamp to hang stationary with respect to the lift) is if there is somehow no tension in the spring to start with.

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Well, of course the real answer is that the people in the elevator would have no time to measure the acceleration of the lamp or the tension on the rope before the elevator hit the bottom of the shaft and they were all killed, so the question is completely moot. :-)

But taking the problem as given: The flaw is in assuming that what is preventing the lamp from falling to the floor of the elevator is the tension on the rope. As others have noted, in reality the tension on the rope is zero.

Suppose that the people in the elevator -- in between screaming and crying -- cut the rope. Would the lamp then fall to the floor? No, because the elevator is accelerating at the same rate as the lamp, so from the point of view of people in the elevator, the lamp would continue to float motionless in the air. Thus, one could easily prove that the tension on the rope is zero by cutting the rope. Or put another way, you could test whether the rope is or is not what is holding the lamp in place by cutting the rope and observing what happens.

Just because something is attached to a rope, one can't jump to the conclusion that the rope is what is holding it in place. If I see a lamp lying on the floor with a rope attached to it, I don't assume that what is preventing the lamp from flying to the far end of the room, or to the ceiling, is tension on the rope!

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Consider a spherical cow, in vacuum....

The lamp will continue to hang from it's rope, which will continue to have tension. Person A will remain on the floor. If Person A stops panicking, reaches up and cuts the rope, the lamp will fall slowly to the floor.

Why? All objects in the lift are pulled downward at g. But the lift cannot freefall at g - the shaft exerts considerable air resistance, the guides and rollers add a bit more (we assume for this exercise the brake has been disabled).

Person A will experience greatly reduced gravity - a small jump may result in them hitting the ceiling of the lift. But they will fall back to the floor. The lamp will appear to rise - this is the tension in the (slightly) elastic rope being released. It will go back down again.

How much residual gravity we have will be nearly impossible to calculate, but 0.1-0.3 g would be reasonable.

Person B will of course see none of this, on account of being on the outside of a rapidly descending steel box.

If this is the Burj Dubai's express lift you just might have time for all these forces to stabilize before person A, the lamp, and the lift come to a messy end in the basement. Text your answer from your phone.

Oh no, not again.

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