I was recently reading through some posts on this site. A question struck me -

Why does an object attached to the end of a string which is further attached to an accelerating car behave differently than it would if the object is attached with a rod welded to the roof?

P.S.- Apparently there is a misunderstanding with my question. What I mean by 'differently' is that the object when attached through a string tries to remain in rest, but then accelerates with the car (inertia), causing the string to make an angle $\theta$ with the vertical, whereas it is not the case when the object is attached through a rod, i.e., the rod always make $0^\circ$ with the vertical.

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Edit - I have added a picture, if that helps...

  • $\begingroup$ Can you clarify what do you mean by 'behave differently'? $\endgroup$ – Cluse Mar 7 at 17:34
  • $\begingroup$ @Tom10 Is it clear now? $\endgroup$ – user290607 Mar 7 at 17:49
  • $\begingroup$ Not to me. I think you're relying on people knowing the class of problem you're talking about, but I don't, so I'll drop it. In general, especially in this type of question with multiple scenarios of things attached together involving angles, etc, a picture really helps. $\endgroup$ – tom10 Mar 7 at 19:07
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    $\begingroup$ Why do you say the rod will always remain vertical? $\endgroup$ – Sandejo Mar 7 at 19:13
  • $\begingroup$ Is the rod able to rotate? Or is it a rod connected to the bus in an unmovable fashion? $\endgroup$ – Deschele Schilder Mar 8 at 9:04

Alright then. Imagine you find yourself on a bus. There is a string hanging from the ceiling. Next to the string, a rod is welded to the ceiling. I think you agree that the string is movable in all directions while the rod is not (unless you attach the rod to a flexible joint).
Now attach a mass to the end of the string and one to the end of the rod. Can you see it? The bus starts to accelerate. The passengers feel themselves pushed to the seat they are in. Just so, both masses feel a force directed to the rear of the bus. The mass on the string is free to move (within the confine the string imposes) and the string will make an angle with the vertical direction. This angle depends on the value of the mass attached. The higher the mass, the smaller the angle (the force of gravity, which pulls the mass down gets bigger). The mass on the rod will not be able to move because the rod is welded to the ceiling. Only if the rod can move in the way the string can move, both the string and the rod will make an angle with the vertical upon acceleration.

Do you get me?

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    $\begingroup$ No. That still does not answer my question. Why does not the rod bend? $\endgroup$ – user290607 Mar 8 at 13:04
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    $\begingroup$ Ah. now its clear. The röd bends a little. If it would bend visibly it wouldnt be a rod. $\endgroup$ – Deschele Schilder Mar 8 at 16:10
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    $\begingroup$ I have accepted your solution. Since the rod bends a little, the horizontal component of the force on the bus is the only force on the object, which is the same as compared to the string. Where is the difference (there is a subtle difference, I think). Please elaborate. $\endgroup$ – user290607 Mar 8 at 17:11
  • $\begingroup$ @Feynstein What do you mean by the force on the bus? $\endgroup$ – Deschele Schilder Mar 8 at 17:14
  • $\begingroup$ The force which causes the bus to accelerate. $\endgroup$ – user290607 Mar 8 at 17:15

I understand I have not understood the question right when I answered the following:

Because of inertia. If no forces are acting on an object, the objects velocity does not change. Using a string the car accelerates the object to a certain velocity and when slowing down the object will continue to move towards the car as no forces are acting on it (until it hits the car where the normal force of the mass of the car slows it down at once). Using a rod the object again accelerates with the car as it is tied to it. But when the car slows down, the normal force of the mass of the car immediately acts on the object thrue the rod, slowing down the object at the same time as the car slows down.

Second Try:

If I understood correctly what is meant here, the reason for the object making an angle to the "vertical" is simply inertia and the fact that it the string has a finite length: The object (attached outside of the car on the ceiling or inside on the ground) wants to stay at rest but is accelerated. The only way to stay as much at rest as possible is to move down (in addition to moving to the opposite side of the acceleration relative to the car) and thus making that angle.

If you meant horizontal instead of "vertical" (the object being attached to the back of the car outside) then the reason is chaos of aerodynamics. Aerodynamics predicts that the object moves in every direction, despite being dragged back by inertia and thus making that angle. But I'm not that familiar with aerodynamics and am therefor not able to give you more details. What I do understand is that when using a fixed rod welded to the car, the object has no degree of freedom whatsoever. Its position is completely determined by the car. A string gives the object all three degrees of freedom $(r,\theta,\varphi)$, inspite of some boundaries: $r\le L$ (length of string), $\varphi\in [0,\pi]$ (object can't go into the car), $\theta\in [0,\pi]$ (boundaries for $\theta$ are normal). Having these degrees of freedom the chaotic consequences of aerodynamics act on the object.

  • $\begingroup$ I think that the object hangs on a string or rod instead of lying on the ground (of the bus or behind it). $\endgroup$ – Deschele Schilder Mar 8 at 9:13
  • $\begingroup$ @DescheleSchilder I have extended my answer. $\endgroup$ – Roger Mar 8 at 11:37
  • $\begingroup$ Do you envision the object in or outside the car? I'm not sure if I understand the connection between the deviation from the vertical and chaos of aerodynamics. If I hang (inside a bus) a mass on a string, then on acceleration of the bus the object the string will deviate from hanging vertical (which a rod melted to the ceiling can't do). $\endgroup$ – Deschele Schilder Mar 8 at 11:45
  • $\begingroup$ @DescheleSchilder I have mistaken vertical with horizontal. The answer is updated now. $\endgroup$ – Roger Mar 8 at 11:51
  • $\begingroup$ @Roger I don't think my question has anything to do with aerodynamics. You can think of the bus in vacuum if you wish. $\endgroup$ – user290607 Mar 8 at 12:34