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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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How is your trigonometry? – ja72 Oct 5 '12 at 12:54
alright i guess. im assuming thats a hint at what i should be looking at? – sri Oct 5 '12 at 12:56
check out my answer, feel free to ask if you don't understand. – Mew Oct 5 '12 at 13:10
up vote 2 down vote accepted

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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Perfect. Thanks a ton! – sri Oct 5 '12 at 13:14

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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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. – sri Oct 5 '12 at 12:53
Note for context: I changed the angle from 45 to 30. – sri Oct 5 '12 at 12:58
+ @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. – Mike Dunlavey Oct 5 '12 at 13:38

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 $a$


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