364146
A small body of mass 10 gram is making harmonic oscillations along a straight line with a time period of \(\pi / 4\) and the maximum displacement is \(10\;cm\). The energy of oscillation is
364147
The total energy of the body executing simple harmonic motion is \(E\) then, the kinetic energy when the displacement is at the mid point of mean and extreme position is
1 \(\dfrac{E}{4}\)
2 \(\dfrac{E}{2}\)
3 \(\dfrac{\sqrt{3} E}{4}\)
4 \(\dfrac{3 E}{4}\)
Explanation:
Total energy of SHM, \(E=\dfrac{1}{2} m \omega^{2} a^{2},(\text { where }, a=\text { amplitude })\) Kinetic energy, \(K=\dfrac{1}{2} m \omega^{2}\left(a^{2}-y^{2}\right) \Rightarrow K=E-\dfrac{1}{2} m \omega^{2} y^{2}\) When \(y=\dfrac{a}{2} \Rightarrow K=E-\dfrac{1}{2} m \omega^{2}\left(\dfrac{a^{2}}{4}\right)=E-\dfrac{E}{4}\) \(K=\dfrac{3 E}{4}\)
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PHXI14:OSCILLATIONS
364146
A small body of mass 10 gram is making harmonic oscillations along a straight line with a time period of \(\pi / 4\) and the maximum displacement is \(10\;cm\). The energy of oscillation is
364147
The total energy of the body executing simple harmonic motion is \(E\) then, the kinetic energy when the displacement is at the mid point of mean and extreme position is
1 \(\dfrac{E}{4}\)
2 \(\dfrac{E}{2}\)
3 \(\dfrac{\sqrt{3} E}{4}\)
4 \(\dfrac{3 E}{4}\)
Explanation:
Total energy of SHM, \(E=\dfrac{1}{2} m \omega^{2} a^{2},(\text { where }, a=\text { amplitude })\) Kinetic energy, \(K=\dfrac{1}{2} m \omega^{2}\left(a^{2}-y^{2}\right) \Rightarrow K=E-\dfrac{1}{2} m \omega^{2} y^{2}\) When \(y=\dfrac{a}{2} \Rightarrow K=E-\dfrac{1}{2} m \omega^{2}\left(\dfrac{a^{2}}{4}\right)=E-\dfrac{E}{4}\) \(K=\dfrac{3 E}{4}\)