A pendulum is hanging from the ceiling of a car having an acceleration `a_(0)` with to the road. Find the angle made by the string with the verical and tension in the string.

Explanation

Let `theta` be an angle made by the string with vertical inertial frame of reference (ground)`:`

`uarr:Tcostheta=mgrArrT=(mg)/(costheta)`
`rarr:Tsintheta=ma_(0)`
`(mg)/(costheta)sintheta=ma_(0)`
`tantheta=(a_(0))/(g)`
`theta=tan^(-1)((a_(0))/(g))`
`T=sqrt((Tsintheta)^(2)+(Tcostheta)^(2)`
`=sqrt((ma_(0))^(2)+(mg)^(2)`
`=m(a_(0)^(2)+g^(2))^(1/2)`
OR
Non`-`inertial frame of reference (car)`:`
If we work from the car frame (an accelerated frame), we have to apply a psedo force `ma_(0)` on the ball towards left.

With respect to the car, bob is in equilibrium.
`Tcostheta-mg=0rArrTcostheta=mg`
`Tsintheta-ma_(0)=0rArrTsintheta=ma_(0)`
`theta=tan^(-1)(a_(0)/(g))`
`T=msqrt((g)^(2)+a_(0)^(2))`