A truck is stationary and has a bob suspended by a light string in a frame attached to the truck. The truck su

Question

A truck is stationary and has a bob suspended by a light string in a frame attached to the truck. The truck suddenly moves to the right with an acceleration of $a$. In the frame of the truck, the pendulum will tilt:

Accepted Answer

1. **Frame of Reference:** The problem asks for the behavior in the frame of the truck. Since the truck is accelerating to the right with acceleration $a$, it constitutes a non-inertial frame of reference.
2. **Pseudo Force:** In a non-inertial frame accelerating with $\vec{a}$, a pseudo force (fictitious force) acts on a particle of mass $m$ given by $\vec{F}_{pseudo} = -m\vec{a}$. Since the truck accelerates to the right, the pseudo force acts on the bob to the **left** with magnitude $ma$.
3. **Forces on the Bob:**
   - Weight ($mg$) acting vertically downwards.
   - Tension ($T$) acting along the string.
   - Pseudo force ($ma$) acting horizontally to the left.
4. **Equilibrium:** The bob comes to equilibrium at an angle $	heta$ with the vertical where the net force is zero. Resolving forces:
   - Horizontal: $T \sin 	heta = ma$
   - Vertical: $T \cos 	heta = mg$
5. **Calculating Angle:** Dividing the horizontal equation by the vertical equation:
   $$ \frac{T \sin 	heta}{T \cos 	heta} = \frac{ma}{mg} $$
   $$ 	an 	heta = \frac{a}{g} \implies 	heta = 	an^{-1}\left(\frac{a}{g}ight) $$
   Thus, the pendulum tilts to the left with an angle $	an^{-1}(a/g)$.
   (Reference: NCERT Class 11, Physics Part I, Chapter 5: Laws of Motion, concepts of equilibrium and non-inertial frames).

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