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In the previous articles, simple harmonic motion was considered from the point of view of the forces that act on the oscillating system. In this article, the kinetic and potential energies in simple harmonic motion are considered. Indeed, the main features of simple harmonic motion can be deduced from energy considerations alone. |
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In the previous section, we considered the simple harmonic motion of a block of mass m, and we showed that if the block is initially stationary and at the far end of its motion, then the displacement x of the block as a function of time is given by
We can now investigate the kinetic and potential energies of the block during its motion. The kinetic energy T is given by
The potential energy of the block for a displacement x can be obtained by calculating the work required to move the mass from zero displacement to displacement x. This is given by
Hence the potential energy V is given by
where in the last step we have noted that
We notice the following characteristics of the kinetic and potential energies of the block. The kinetic energy is a maximum when
and from the expression for x above, it can be seen that x =0 when the kinetic energy is a maximum. The potential energy is zero at these points.
Similarly, the potential energy is a maximum when
From the expression for x above, it can be seen that the block is at the points of maximum displacement, with the kinetic energy being zero at these points.
The total energy E is just the sum of the kinetic and potential energies:
Thus the total energy is constant – as the block undergoes simple harmonic motion, there is a continual exchange between the kinetic and potential energy. The total energy is equal to the maximum values of the kinetic and potential energies. This is because, as we discussed above, the total energy is constant and at the point of maximum kinetic energy the potential energy is zero, and vice versa.
Consideration of the constant total energy E provides an alternative means of deriving the motion of a simple harmonic oscillator. The total energy can be written as
From which simple re-arrangement gives
This equation can be solved to give the same sinusoidal variation of x with time that we saw in the previous section. It can be seen from this expression that the displacement x is limited such that
Otherwise, the square root in the governing equation is not a real number. The magnitude of the total energy E therefore limits the amplitude of the simple harmonic motion. Energy arguments of this type are often used in quantum mechanical treatments of oscillating and other mechanical systems.