Question

When the kinetic energy of a body executing S.H.M. is $1 / 3$ of the potential energy. The displacement of the body is $x$ percent of the amplitude, where $x$ is :

• A. 33
• B. 87
• C. 67
• D. 50

Answer 1

Answer: (B)

The relation for kinetic energy of S.H.M. is given by
$=\frac{1}{2} m \omega^{2}\left(a^{2}-y^{2}\right)$
Potential energy is given by
$=\frac{1}{2} m \omega^{2} y^{2}$
Now for the condition of question and from eqs. (1) and (2)
$\frac{1}{2} m \omega^{2}\left(a^{2}-y^{2}\right)=\frac{1}{3} \times \frac{1}{2} m \omega^{2} y^{2}$
or $\frac{4}{6} m \omega^{2} y^{2}=\frac{1}{2} m \omega^{2} a^{2}$
or $y^{2}=\frac{3}{4} a^{2}$
So, $y =\frac{a}{2} \sqrt{3}=0.866\, a$
$\approx 87 \%$ of amplitude