Question

A particle is vibrating in SHM with an amplitude of 10 cm. What fraction
of the total energy is kinetic when the displacement of the particle from the
mean position is 5 cm? At what displacement is the energy half kinetic and
half potential,?
​

Answer 1

Kinetic energy is 0.75th part of the total energy. At displacement = 7.07 cm, the energy is half kinetic and half potential.

Here, according to the given information, we are given that,

A particle is vibrating in SHM with an amplitude of 10 cm.

Now, we know that, the Kinetic energy when a particle is vibrating in simple harmonic motion, is given as,

$\frac{1}{2} m\beta ^{2}(a^{2}-x^{2} )$ .

Also, the potential energy is given as,

$\frac{1}{2} m\beta ^{2} x^{2}$.

Now, $\beta$ is equal to ω here.

Now, it is given that, the amplitude = 10 cm.

Also, the total energy is the sum of the potential energy and the kinetic energy.

Then, Total energy = $\frac{1}{2} m\beta ^{2} a^{2}$.

Now, let kinetic energy be yth part of the total energy.

Now, y($\frac{1}{2} m\beta ^{2} a^{2}$) = $\frac{1}{2} m\beta ^{2}(a^{2}-x^{2} )$

Or, $ya^{2} =a^{2} -x^{2}$

Here, x = 5cm.

then, we get,

$100(1-y) = 25$

Or, 1-y = 0.25

Or, y = 0.75

Again, we have, $a^{2} -x^{2} =x^{2}$

Or, 100 = 2$x^{2}$

Or, $x=\sqrt{50}$ = 7.07 cm.

Hence, kinetic energy is 0.75th part of the total energy. At displacement = 7.07 cm, the energy is half kinetic and half potential.

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