Calculating car's acceleration from change in angle of hanging object?

Question

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.

Answer 1

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 $a = g \tan \theta$

Answer 2

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.

Answer 3

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 $a$