First slide

Simple harmonic motion

Question

The total energy of a vibrating particle in S.H.M is E. If it’s amplitude is tripled and time period are doubled, it’s total energy will be

Moderate

A

16 E
B

8 E
C

4 E
D

$\frac{9}{4}$ E

Solution

As $E= \frac{1}{2} {mw}^{2} A^{2}$ , mass of oscillator=m, angular velocity=w, Amplitude=A
$E= \frac{1}{2} m {[\frac{2π}{T}]}^{2} A^{2}$ -----(1)
As new time period $T^{1} = 2 T$ , new amplitude $A^{1} = 3 A$ , substitute in new energy $E^{1}$
$∴ E^{1} = \frac{1}{2} m {[\frac{2 π}{2 T}]}^{2} \times {(3 A)}^{2}$
$E^{1} = \frac{1}{2} m \times \frac{1}{4} {[\frac{2π}{T}]}^{2} \times 9 A^{2}$ ------(2)
From (1) and (2), $E^{1} = \frac{9}{4} E$

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