First slide

Applications of SHM

Question

A liquid of density $ρ$ contained in a vertical U tube filled to a height h on each limb. It executes simple harmonic motion in its limbs when a small disturbance is given. Then the time period of its oscillation is

Moderate

A

$T = 2 π \sqrt{\frac{h}{g}}$
B

$T = 2 π \sqrt{\frac{2 h}{g}}$
C

$T = 2 π \sqrt{\frac{g}{h}}$
D

$T = 2 π \sqrt{\frac{g}{2 h}}$

Solution

The restoring force responsible for oscillation is the weight of the liquid displaced. Hence using the condition for simple harmonic motion we can say that $a = - ω^{2} y$ . Here $a = \frac{F}{m}$ . If the mass per unit length is $μ$ , then $F = μ (2 y) g$ and $m = μ (2 h)$ . Simplifying we get $a = \frac{μ (2 y) g}{μ (2 h)} = \frac{y g}{h}$
Comparing this with $a = - ω^{2} y,$ we get
$ω^{2} = \frac{g}{h} .$
where $ω = \frac{2 π}{T}$ . Hence the time period of oscillation is $T = 2 π \sqrt{\frac{h}{g}}$

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