A simple pendulum of length l is suspended from the ceiling of a trolley which is moving, without friction, down an inclined plane of inclination $\alpha$. The time period of the pendulum is given by $T=2\pi \sqrt{\frac{l}{g_{\text{eff }}}}, where g_{\text{eff }}$ is given by

The acceleration of the vehicle down the plane is $g\sin \alpha$. The reaction force acting on the pendulum bob gives it an acceleration $a=g\sin \alpha$ up the plane. This acceleration has two rectangular components $a_x=a\cos \alpha =g\sin \alpha \cos \alpha$ and ${\alpha }_y=a\sin \alpha =g{\sin }^2\alpha$ as shown in Fig. 

The effective acceleration due to gravity acting on the bob is given by
$$\begin{gathered} g_{\text{eff }}^2=a_x^2+{\left(g-a_y\right)}^2 \\ =a_x^2+g^2+a_y^2-2g{a}_y \\ =g^2{\sin }^2\alpha {\cos }^2\alpha +g^2+g^2{\sin }^4\alpha \\ -2g^2{\sin }^2\alpha \\ =g^2{\sin }^2\alpha \left({\cos }^2\alpha +{\sin }^2\alpha \right)+g^2 \\ -2g^2{\sin }^2\alpha \\ =g^2\left(1-{\sin }^2\alpha \right)=g^2{\cos }^2\alpha \\ \text{ or } g_{\text{eff }}=g\cos \alpha \\ \text{ Now } T=2\pi \sqrt{\frac{L}{g_{\text{eff }}}}=2\pi \sqrt{\frac{L}{g\cos \alpha }} \end{gathered}$$