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The question essentially is based on a situation like this-

A car has a small object hung from the cieling on a string (apparently at an angle of 0 degrees to the ceiling).

The car is accelerating and the object is now hanging at a 30 degree angle (to the ceiling). How would I figure out how much the car is accelerating.

PS - This is homework but Im stuck and would appreciate any advice. Thanks.

Edit: changed angle from 45 to 30.

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  • $\begingroup$ How is your trigonometry? $\endgroup$ – ja72 Oct 5 '12 at 12:54
  • $\begingroup$ alright i guess. im assuming thats a hint at what i should be looking at? $\endgroup$ – sri Oct 5 '12 at 12:56
  • $\begingroup$ check out my answer, feel free to ask if you don't understand. $\endgroup$ – Kenshin Oct 5 '12 at 13:10
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This problem can be tackled using the equivalence principle. This basically means that the accelerating car can be thought of from the perspective of the hanging object, as a horizontal gravitational field, with an acceleration equal to that of the car.

Therefore we effectively have two forces acting on the object. One downwards of $mg$, the other horizontally of $ma$, where a is the acceleration of the car.

The angle of the resultant force is given by $\tan \theta = \frac{m a}{m g} = \frac{a}{g}$

Therefore $a = g \tan \theta$

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  • $\begingroup$ Won't the angle also depend on the density of the air and the object? $\endgroup$ – Aniansh Nov 17 '16 at 0:28
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    $\begingroup$ @Aniansh so long as the object is denser than air it won't. $\endgroup$ – Kenshin Nov 17 '16 at 7:52
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This is a neat example because the object makes its own force triangle - it's being pulled down by gravity and sideways by the car's acceleration. And the 45° angle means that the forces are equal.

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  • $\begingroup$ Ahh I see. The real question has a 13 degree angle. I get it when they are equal, but how would I forumalte the link between the force-acceleration and force-gravity now. $\endgroup$ – sri Oct 5 '12 at 12:53
  • $\begingroup$ Note for context: I changed the angle from 45 to 30. $\endgroup$ – sri Oct 5 '12 at 12:58
  • $\begingroup$ + @sri: Just make a right triangle by drawing a vertical line from the attachment point, and a horizontal line from the object. Measure the height h and base b of the right triangle. Then just do the proportion. Acceleration a is to gravity g as b is to h. $\endgroup$ – Mike Dunlavey Oct 5 '12 at 13:38
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D'Alembert's principle can be used to convert any dynamic system into a static one by converting accelerations into equal but opposite inertia forces.

Hence in the diagram below there is a horizontal force acting to the right with value of $m a$. Now all you have to do is the force balance in x-axis and y-axis to get $T$ and $a$

FBD

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