The potential energy of a simple harmonic oscillator, when the particle is half way to its end point is (where E is the total energy)

Question

The potential energy of a simple harmonic oscillator, when the particle is half way to its end point is (where E is the total energy) (A)  (1/4)E  (B)  (1/2)E  (C)  (2/3)E  (D)  (1/8)E

Tardigrade Admin · Accepted Answer

Potential energy of a simple harmonic oscillator

U = \frac{1}{2} m ω^{2} y^{2}

Kinetic energy of a simple harmonic oscillator

K = \frac{1}{2} m ω^{2} (A^{2} - y^{2})

Here

y =

displacement from mean position

A =

maximum displacement (or amplitude) from mean position
Total energy is

E = U + K

= \frac{1}{2} m ω^{2} y^{2} + \frac{1}{2} m ω^{2} (A^{2} - y^{2})

= \frac{1}{2} m ω^{2} A^{2}

When the particle is half way to its end point ie, at half of its amplitude then

y = \frac{A}{2}

Hence, potential energy

U = \frac{1}{2} m ω^{2} (\frac{A}{2})^{2}

= \frac{1}{4} (\frac{1}{2} m ω^{2} A^{2})

U = \frac{E}{4}

Q. The potential energy of a simple harmonic oscillator, when the particle is half way to its end point is
(where $E$ is the total energy)

$\frac{1}{4} E$

$\frac{1}{2} E$

$\frac{2}{3} E$

$\frac{1}{8} E$

Q. The potential energy of a simple harmonic oscillator, when the particle is half way to its end point is (where E is the total energy)

41​E

21​E

32​E

81​E

Q. The potential energy of a simple harmonic oscillator, when the particle is half way to its end point is
(where $E$ is the total energy)

$\frac{1}{4} E$

$\frac{1}{2} E$

$\frac{2}{3} E$

$\frac{1}{8} E$