First slide

Circular motion in horizontal plane

Question

A small sphere is suspended by a string of length $\sqrt{3}$ m. If the sphere rotates in a horizontal circle of radius 2m with uniform speed, the time period of rotation of the sphere is

Moderate

A

$\frac{π}{\sqrt{5}} \sec$
B

$\frac{2 π}{\sqrt{10}} \sec$
C

$\frac{2 π}{\sqrt{5}} \sec$
D

$\frac{π}{\sqrt{10}} \sec$

Solution

$T \sin θ = m ω^{2} r = m ω^{2} (l \sin θ)$
$\Rightarrow T = m ω^{2} l ....... (1)$
Also, $T \cos θ = m g ....... (2)$
$∴ ω^{2} l \cos θ = g \Rightarrow ω = \sqrt{\frac{g}{l \cos θ}}$
$∴$ Time period of rotation, $T^{1} = \frac{2 π}{10} = 2 π \sqrt{\frac{2 \times \cos 60^{o}}{10}} \sec = 2 π \sqrt{\frac{1}{10}} \sec$

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