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Let's say we are traveling in a train. The path has two parts: one at ground-level and the other moving up on the ramp. The ramp has an inclination of $\arctan\frac{a}{g}$ with the horizontal, where $a$ is the acceleration of the train on level ground and $g$ is the acceleration due to gravity.

The train does not accelerate on the ramp, but moves with a constant velocity. Can we comment where we are (sitting inside the train of course!) when we have only a pendulum hanging on the roof to observe. (windows are blackened)

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    $\begingroup$ generally, on this site "homework-type" questions are tagged as homework even if they don't arise from actual homework assignments. But I'm not the expert on this, I was just trying to be helpful. $\endgroup$
    – N. Virgo
    Commented Mar 19, 2013 at 4:56
  • $\begingroup$ en.wikipedia.org/wiki/Equivalence_principle $\endgroup$
    – DilithiumMatrix
    Commented Mar 19, 2013 at 5:11
  • $\begingroup$ @zhermes What i do with Equivalence-principle? $\endgroup$
    – ABC
    Commented Mar 19, 2013 at 6:10
  • $\begingroup$ Hi exploringnet. If you haven't already done so, please take a minute to read the definition of when to use the homework tag, and the Phys.SE policy for homework-like problems. $\endgroup$
    – Qmechanic ♦
    Commented Mar 19, 2013 at 8:20

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If the train is an ideal isolated system, the answer is no, we can't tell where we are. As zhermes pointed out, the equivalence principle states that the gravitational force experienced by a body at rest or moving with a constant velocity is the same as the pseudo-force experienced by that body is a non-inertial frame of reference. This simply states that the inertial mass is equal to the gravitational mass. To see that this is so, instead of thinking that the train is accelerating, imagine that you place another gravitational field that pulls on the pendulum in the same direction as the inertial force. Also, read about Einstein's thought experiment involving his famous elevator. In some ways its similar to your question.

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Although the gravitational/inertial force causing the pendulum to tilt in the same way in both cases (see Equivalence principle link of @zhermes) thus not allowing you to see whether you are on accelerating or on the slope, you will probably be able to feel the difference.

The reason is that although the force parallel to the train is equal (which causes the pendulum to behave in the same way), the normal force acting on you is not so depending on the value of $\arctan\left(\frac{a}{g}\right)$ (whether it is sufficiently larger than 0) you will probably be able to feel whether you are on the incline or not.

[ADDITION after discussion with @markovchain]

Thinking along the line of parallel and normal forces I have to expand my answer a bit: a RIGID pendulum will look exactly the same on the flat and sloped part. A FLEXIBLE pendulum however, will be straight when moving up the slope, but curved when accelerated on the horizontal surface because it will have both a component pulling it to the left and a component (gravity) pulling it down.

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  • $\begingroup$ But the question was, can we do it by observing the pendulum? That is, will it "tilt" towards one side? I think it's the same thing as asking, if we have water in a glass and the train goes up the ramp, will the water level, from my perspective, tilt? $\endgroup$
    – markovchain
    Commented Mar 21, 2013 at 7:51
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    $\begingroup$ No, the question was: "Can we comment where we are (sitting inside the train of course!) when we have only a pendulum hanging on the roof to observe". I understood that as: the only thing we have to observe is the pendulum, we can't see anything else. He did not mention anything about not having a feeling (in fact he specifically mentioned that we are inside the train) $\endgroup$
    – Michiel
    Commented Mar 21, 2013 at 8:00
  • $\begingroup$ Why would he mention the pendulum if he didn't want to know about it? Disregarding that is functionally the same as "can we comment where we are... if I'm eating lunch too?" The fact you eat lunch doesnt have to do with anything -- but a pendulum might. Wouldn't that be why it's mentioned in the question? I'm not disputing your answer. You're quite correct. But isn't it interesting to wonder if the pendulum would tilt to one side when the train gets on the ramp? $\endgroup$
    – markovchain
    Commented Mar 21, 2013 at 8:06
  • $\begingroup$ I am not disregarding the pendulum. I am just saying that the pendulum, by the equivalance principle that zhermes mentioned, will not give us any information because it will hang under the same angle with respect to the train. However, I continue my answer by explaining that not all forces are equal (the normal force is not) so that could give you information on where you are. $\endgroup$
    – Michiel
    Commented Mar 21, 2013 at 8:52

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