Question

A metal sphere is hanging from the ceiling of a vehicle. If the vehicle is moving along a horizontal road with uniform acceleration \(a\), then the suspended thread of the sphere gets inclined to the vertical at an angle \(\theta\). The value of acceleration \(a\) is (\(g\) = acceleration due to gravity)

• A. \(g \cos\theta\)
• B. \(g\)
• C. \(g \sin\theta\)
• D. \(g \tan\theta\)

Hint:

Inclination of a suspended body in an accelerating frame is governed by pseudo force.

Answer 1

The Correct Option is D

Step 1: Forces acting on the sphere.
Weight \(mg\) acts vertically downward and pseudo force \(ma\) acts horizontally opposite to acceleration.
Step 2: Resolving tension components.
\[ T\cos\theta = mg \] \[ T\sin\theta = ma \]
Step 3: Taking ratio of equations.
\[ \tan\theta = \frac{a}{g} \]
Step 4: Solving for acceleration.
\[ a = g\tan\theta \]
Step 5: Conclusion.
The acceleration of the vehicle is \(g\tan\theta\).