Question

What will be the time period of the pendulum of length l suspended from the ceiling of a cart which is sliding without friction on an inclined plane of inclination θ ?

• A. $2 \text{\pi } \sqrt{\frac{ l }{\text{g} \text{cos} \text{\theta }}}$
• B. $- 2 \text{\pi } \sqrt{\frac{3 l ⁡}{\text{g} \text{cos} \text{\theta }}}$
• C. $4 \text{\pi } \sqrt{\frac{2 l ⁡}{\text{g} \text{cos} \text{\theta }}}$
• D. $- 3 \text{\pi } \sqrt{\frac{4 l ⁡}{\text{2g} \text{cos} \text{\theta }}}$

Answer 1

Answer: (A)

Here, point of suspension has an acceleration. $\overset{ \rightarrow }{\text{a}}$ = g sin θ (down the plane). Further, $\overset{ \rightarrow }{\text{g}}$ can be resolved into two components g sin θ (along the plane) and g cos θ (perpendicular to plane).

$\overset{ \rightarrow }{\text{g}}_{\text{eff}} = \overset{ \rightarrow }{\text{g}} - \overset{ \rightarrow }{\text{a}}$
= g cosθ (perpendicular to plane)
$\text{T} = 2 \text{\pi } \sqrt{\frac{ l }{\left|\right. \overset{ \rightarrow }{\text{g}}_{\text{eff}} \left|\right.}}$
$= 2 \text{\pi } \sqrt{\frac{ l }{\text{g} \text{cos} \text{\theta }}}$