First slide

Simple hormonic motion

Question

The bob of a simple pendulum executes simple harmonic motion in water with a period t, while the period of oscillation of the bob is $t_{0}$ in air. Neglecting frictional force of water and given that the density of the bob is $(\frac{4}{3}) \times 1000 kg / m^{3}$ . Which relationship between $t and t_{0}$ is true?

Moderate

A

$t = t_{0}$
B

$t = \frac{t_{0}}{2}$
C

$t = 2 t_{0}$
D

$t = 4 t_{0}$

Solution

$∴ t_{0} = 2 π \sqrt{\frac{l}{g}}$
Effective weight of bob inside water,
$W^{'} = mg - thrrust = Vρg - {Vρ}^{`} g$
$\Rightarrow {Vρg}_{eff} = V (ρ - ρ^{1}) g, where, ρ = Density of bob$
$\Rightarrow g_{e f f} = (1 - \frac{ρ^{'}}{ρ}) g a n d ρ^{'} = D e n s i t y o f w a t e r$
$∴ t = 2 π \sqrt{\frac{l}{g_{eff}}} = 2 π \sqrt{\frac{l}{(1 - \frac{ρ^{'}}{ρ}) g}}$
$∴ \frac{t}{t_{0}} = \sqrt{\frac{1}{1 - \frac{ρ^{'}}{ρ}}} = \sqrt{\frac{1}{1 - \frac{3}{4}}}$
$(∵ ρ^{'} = 10^{3} kg / m^{3} and ρ = \frac{4}{3} \times 10^{3} kg / m^{3})$
$\Rightarrow t = 2 t_{0}$

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