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| Energy in SHM |
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| To cause a displacement of a body
controlled by elastic forces, work has to be done. For example, work has to
be done in stretching a spring against the elastic forces. This work will be
stored in the body as potential energy. When the deforming force is
released, this potential energy manifests in the form of kinetic energy
which makes the body to move. Thus, a body in SHM, has both potential energy
and kinetic energy. |
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The work done in displacing a body through a distance y is given by force X displacement. In SHM, the force is not a constant but
proportional to the displacement. Therefore, an average force  can be assumed to produce a displacement y. |
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| But F = Ky in magnitude |
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| Thus, the total energy is a constant. |
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| At the equilibrium position, y = 0 |
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| At the extreme position, y = A, |
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| As the body oscillates, there is a
transformation between kinetic energy and potential energy, maintaining the
total energy constant. The loss in potential energy appears as a gain of
kinetic energy and vice versa. |
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| The constant interchange of energy between potential and kinetic forms is essential for producing and maintaining any type of oscillation. In the case of the oscillating bob of a simple pendulum, it loses kinetic energy after passing through the middle of the swing and then stores the energy as potential energy, as it rises to the top of the swing. The reverse occurs as it swings back. In the case of oscillating layers of air, when a sound wave passes, kinetic energy of the moving air molecules is converted to potential energy when the air is compressed. In the case of electrical oscillations a coil L and a capacitor C in the circuit constantly exchange energy, this is stored alternately in the magnetic field of L and the electric field of C. |
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| The figure shows graphs of kinetic energy
and potential energy versus displacement. Each of them is a parabola. The
sum of the potential energy and the kinetic energy values remains constant
as represented by the horizontal line. At the points where the two curves
cross, we can write |
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| Equation (2.53) can also be written as |
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| Thus, the energy of a body executing SHM is directly proportional to |
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 the mass of the body |
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 the square of the frequency |
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 the square of the amplitude |
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