Question

A pendulum of mass m as hanging from the ceiling of a car having an acceleration $a_0$ with respect to the road. Find the angle made by the string with the vertical.

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Mass of the bob is m and the string makes an angle $\theta$ with the vertical. We shall work from the car frame. This frame is non-inertial as it has an acceleration with respect to an inertial frame (the road). Hence, if we use Newton's second law we have to include a pseudo force.

The free body diagram is shown in figure. As the bob is at rest (remember we are discussing the motion with respect to the car) the force in (a), (b) and (c) should add to zero. Take X-axis along the forward horizontal direction and Y-axis along the upward vertical direction. The components of the forces along the X-axis give

$Tsin\theta -m{a}_0=0$ or, $Tsin\theta =m{a}_0$ ----------(i)

And the components along the Y-axis give

$Tcos\theta -mg=0$ or, $Tcos\theta =mg$ ----------(ii)

Dividing (i) by (ii) $tan\theta =\frac{a_0}{g}$

Thus, the string makes an angle $ta{n}^{-1}\left(\frac{a_0}{g}\right)$ with thw vertical.