Q. 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 ?

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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

$- 3 \text{\pi } \sqrt{\frac{4 l ⁡}{\text{2g} \text{cos} \text{\theta }}}$

Solution:

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)
$\text{T} = 2 \text{\pi } \sqrt{\frac{ l }{\left|\right. \overset{ \rightarrow }{\text{g}}_{\text{eff}} \left|\right.}}$
$= 2 \text{\pi } \sqrt{\frac{ l }{\text{g} \text{cos} \text{\theta }}}$

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