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enter image description here

When the train accelerates, the bob would also accelerate with the same magnitude and direction as the train. From the free body diagram, only tension and weight are exerted on the bob. I understand how to relate the "horizontal component" of the tension to the acceleration of the bob. It moves "backward" and in fact its vertical position is different from when the train is not accelerating.

I am not sure whether inertia could explain this situation. What is the force moving it backward/ lifting with the height?

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    $\begingroup$ This video might help. $\endgroup$ – user238497 Nov 20 '19 at 15:40
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    $\begingroup$ Couldn't be easier, the hinge at the top is being pushed forward - it's that simple. $\endgroup$ – Fattie Nov 21 '19 at 0:32
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    $\begingroup$ It's the same force that is accelerating the train forward, but viewed from the relative perspective of the accelerating train itself. $\endgroup$ – RBarryYoung Nov 21 '19 at 15:52
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    $\begingroup$ Suppose the train is moving due east, and is accelerating in the eastward direction. In what sense is the bob moving backwards? The bob is also moving east; it is certainly not getting more and more westward. Can you clarify the question? $\endgroup$ – Eric Lippert Nov 21 '19 at 18:05
  • $\begingroup$ @Eric You're using an inertial frame there, The OP is using an non-inertial frame. If you are seated in the train, faicing forward such that (with the train at rest) the bob hangs by your shoulder, then when the train begins accelerating forward, the bob will swing to a location behind your shoulder (while, as you say, moving in the train's forward direction, just slower at first than you move in that direction). $\endgroup$ – dmckee --- ex-moderator kitten Nov 21 '19 at 20:11
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The bob doesn't move backwards at all. The train is moving forwards (according to your reference frame) and if the bob wasn't attached to the train it would remain stationary. Since it is attached to the train there is a tension in the wire (as the top anchor point moves with train and the bottom is attached to the weight, so stretching the wire slightly). The vertical component of that tension (which is in the direction of the wire) counteracts gravity and allows the bob to rise whereas the horizontal component accelerates the weight in the direction of the train.

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  • $\begingroup$ As the bob rises when the train accelerates, does it mean that the vertical component of the tension is greater than the weight of the bob? And is it related to inertia? Thanks. $\endgroup$ – mckong Nov 20 '19 at 15:48
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    $\begingroup$ @James surely for the bob to move upwards (as it swings back) the vertical component of tension must be temporarily higher than mg. Though once in a steady state it will be the same. $\endgroup$ – rghome Nov 20 '19 at 17:01
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    $\begingroup$ @rghome: My apologies. I was discussing steady state and failed to realize that you were discussing the dynamics. $\endgroup$ – James Nov 20 '19 at 18:03
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    $\begingroup$ @mckong Pseudo-forces only arise in non-inertial frames. Since we are analyzing the situation from an inertial reference frame (well, inertial enough for this situation), there are none. If you analyze the situation from the train's reference frame, then there will be a pseudo-force. $\endgroup$ – user253751 Nov 20 '19 at 23:27
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    $\begingroup$ @mckong it's very important to know not only what reference frame is being used but what moment in time is being depicted. The bob isn't rising at all now because this is a snap shot of a moment when the bob has stopped moving up or down. This is counter intuitive because it means the train has been constantly accelerating for an unusually long time to give the bob time to settle down. Try this experiment on a real train and you'll see a fair bit of swinging unless you put a motion damper on the bob. Here's a video of a similar experiment. $\endgroup$ – candied_orange Nov 22 '19 at 19:33
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The answer by rghome is correct when Earth's surface is chosen as the reference frame.

Now consider this with the train car as the reference frame.

According to the Equivalence Principle, the effect of an accelerating frame of reference is identical to a gravitational field.

This means that if you are in the train car (without windows), you have no way of knowing whether...

  1. The train car is accelerating, or
  2. gravity has increased and the angle of gravitational field has shifted. In this new gravitational field, the bob is still hanging "straight" and the train car floor is no longer perpendicular to the direction of gravity.

If you were standing in the accelerating train car, it would likewise be hard to distinguish whether...

  1. the car was accelerating, or
  2. one end of the car had been elevated.
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    $\begingroup$ You don't need to appeal to Einstein except to find a decent name for the phenomena. At the level of this problem (rather than related to light and time) all this was well understood in the context of inertial pseudo-forces in classical mechanics. The big issue is that the pseudo-force associated with straight-line accelerations is the only one that doesn't have a name. $\endgroup$ – dmckee --- ex-moderator kitten Nov 20 '19 at 18:48
  • $\begingroup$ Could this pseudo force be calculated? And what is exerting this force to the bob? $\endgroup$ – mckong Nov 20 '19 at 22:16
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    $\begingroup$ @mckong: The value of the horizontal "pseudo" force is equal to the bob mass times the horizontal acceleration ($ma$). In my answer I was trying to explain that from an observer's point of view from inside the train, the train's acceleration toward the left side of your diagram is indistinguishable from a gravitational pull of a massive object on the right side of the train. That's why the equation for the pseudo-force ($ma$) looks so much like the force due to gravity ($mg$). $\endgroup$ – James Nov 20 '19 at 22:52
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    $\begingroup$ @dmckee: I didn't intend to complicate the question by invoking the equivalence principle. To me, the addition of a second gravitational pull just seems much easier to grasp than having to imagine acceleration and pseudo forces. $\endgroup$ – James Nov 21 '19 at 19:59
  • $\begingroup$ @dmckee Just for the record, I have sometimes heard it called "the" inertial force. $\endgroup$ – Javier Nov 22 '19 at 19:46
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This is the problem of accelerating reference frames. Newton's first law:

In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force

can be seen as a condition for the other laws to hold true. The frame that is moving with the train is obviously an accelerating reference frame, so it is not inertial and we can expect forces where there shouldn't be. These forces that appear in accelerating reference frames are called fictitious forces or pseudo forces. Imagine you are on a very fast merry go round and you place a bucket of water in the center. The water will bulge to the sides of the bucket with no apparent cause (from your frame of reference). This is also an effect of non-inertial frames.

When you do find a frame that is inertial everything works like you expect. Newton's laws hold and there are no fictitious forces. From the frame that is outside the train everything looks fine. The train is accelerating and the bob is pulled along with it, only with delay because the force has to act via the rope. The rope can only exert sideways force if the bob is not hanging directly under it.

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Nothing is moving it backward

The train is accelerating, but the bob is not. In order for the bob to be accelerated at the same rate as the train, there needs be a force acting sideways on it. This is effected as the displacement of $\theta$ on $\vec T$- which combined with gravity $m \vec g$ produces a resultant acceleration $\vec a$

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  • $\begingroup$ This is only true in certain frames of reference. If you're sitting in the train as the engines start working, you clearly see the bob moving backwards. That's not an invalid observation, it's just not an inertial reference frame. $\endgroup$ – Arthur Nov 22 '19 at 8:33
  • $\begingroup$ @Arthur I am not saying it is not moving backwards, just that it is not being moved backwards. It is being accelerated, via a string at an angle, forwards. The resultant virtual image of a bob "hanging" askew is so in order for the forces on it to balance. $\endgroup$ – Stian Yttervik Nov 22 '19 at 9:03
  • $\begingroup$ "The train is accelerating, but the bob is not" and "I'm not saying it is not moving backwards" seem like somewhat contradictory statements to me. I was objecting to the former, but if you don't stand by it any more, then that's fine. Also, according to a passenger's reference frame, the train is stationary, and the ball is being moved backwards, by gravity. $\endgroup$ – Arthur Nov 22 '19 at 9:09
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Gravity!!! I know you might be surprised with this answer, but it is not the good old classic Newton's gravitation. This is general relativity, which basically states that all laws of physics must be the same irrespective of the reference frame, this symmetry also known as equivalence principle, leads to what we think of as centrifugal forces.

The equivalence principle states that gravity is just acceleration, which is exactly what is happening here.

There is a very beautiful explanation given by Steven Weinberg, in a short interview. Link- https://youtu.be/0gSomorLJQU

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