A car is moving on a plane inclined at $30^\circ$ to the horizontal with an acceleration of $10\ \text{m s}^{-2}$ parallel to the plane upward. A bob is suspended by a string from the roof of the car. The angle in degrees which the string makes with the vertical is ______. (Take $g = 10\ \text{m s}^{-2}$)
Numerical value type. Enter your answer.
Answer: 30
In the car's frame the bob feels gravity $g$ downward and a pseudo force $ma$ down the incline. Effective gravity $\vec g_{\text{eff}} = \vec g - \vec a$.
$\vec a = (10\cos30^\circ, 10\sin30^\circ) = (5\sqrt3, 5)$, $\vec g = (0, -10)$.
$\vec g_{\text{eff}} = (-5\sqrt3, -15)$. Angle with the vertical: $\tan\theta = \dfrac{5\sqrt3}{15} = \dfrac{1}{\sqrt3} \Rightarrow \theta = 30^\circ$.
Solution by Sreeraj P, M.Sc Physics