# Getting different answers by different methods for angle made by a pendulum moving with constant acceleration

A point mass $$m$$ is hanging by a string of length $$l$$ in a car moving with a constant acceleration $$a$$. Using car frame and pseudo force, we easily get that the angle made by string with vertical is :

$$\theta = \arctan(\frac{a}{g})$$

However when I solve it using work-energy theorem , I'm getting that :

$$\theta = 2\arctan(\frac{a}{g})$$

Here's how I did it : When the mass $$m$$ will finally make an angle $$\theta$$ which we need to find, it's velocity will be zero ( working from car frame). So change in kinetic energy is zero. Hence, sum of work done by each forces is zero so that :

$$W_g+ W_T+ W_f=0$$

$$W_g$$ is work done by gravity which is $$-mgl(1-\cos(\theta))$$.

$$W_T$$ is work done by tension which is zero because tension force was always perpendicular to displacement vector at any instant.

$$W_f$$ is work done by pseudo force which is equal to $$mal\sin(\theta)$$.

Hence , by work energy theorem :

$$mal\sin(\theta)=mgl(1-\cos(\theta)$$

Which gives ,

$$\theta = 2\arctan(\frac{a}{g})$$

Here's a diagram to help visualise the situation :

Note that vertical displacement is $$l(1-\cos\theta)$$ and $$ma$$ is pseudo force.

Please explain in simple language what wrong is going here. Thanks !