First slide

Circular motion in horizontal plane

Question

Three masses of small size are attached by light inextensible strings of various lengths to a point O on the ceiling. All of the masses swing round
in horizontal circles of various radii with the same angular frequency $ω$ (one such circle is drawn in the shown figure.) Then pick up the correct statement.

Moderate

A

The vertical depth of each mass below point of suspension from ceiling is different.
B

The radius of horizontal circular path of each mass is same.
C

All masses revolve in the same horizontal plane.
D

All the particles must have same mass.

Solution

For conical pendulum of length $λ$ mass m moving along horizontal circle as shown
$T \cos θ = mg$ -------------(i)
$T \sin θ = {mω}^{2} l \sin θ$ --------(ii)
From equation (i) and equation (ii),
$l \cos θ = \frac{g}{ω^{2}}$
$l$ cos $θ$ is the vertical distance of bob below O point of suspension. Hence if $ω$ of all three pendulums are same, they shall revolve in same horizontal plane.

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