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Question
What is meant by simple harmonic oscillation? Give examples and explain why every simple harmonic motion is a periodic motion whereas the converse need not be true.
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Solution
Simple harmonic motion is a special type of oscillatory motion in which the acceleration or force on the particle is directly proportional to its displacement from a fixed point and is always directed towards that fixed point. In one dimensional case, let x be the displacement of the particle and ax be the acceleration of the particle, then
ax ∝ x .................(1)
ax = −bx ...............(2)
where b is a constant which measures acceleration per unit displacement and dimensionally it is equal to T-2.
By multiplying by the mass of the particle on both sides of equation (1) and from Newton’s second law, the force is
Fx = −kx ….........(3)
where k is a force constant which is defined as force per unit length. The negative sign indicates that displacement and force (or acceleration) are in opposite directions. This means that when the displacement of the particle is taken towards the right of equilibrium position (x takes positive value), the force (or acceleration) will point towards equilibrium (towards left) and similarly, when the displacement of the particle is taken towards left of equilibrium position (x takes negative value), the force (or acceleration) will point towards equilibrium (towards right). This type of force is known as restoring force because it always directs the particle executing simple harmonic motion to restore to its original (equilibrium or mean) position. This force (restoring force) is central and attractive whose center of attraction is the equilibrium position.
In order to represent in two or three dimensions, we can write using vector notation
`vec"F" = -"k"vec("r")` ...(4)
where `vec"r"` is the displacement of the particle from the chosen origin. Note that the force and displacement have a linear relationship. This means that the exponent of force `vec"F"` and the exponent of displacement `vec"r"` are unity. The sketch between cause (magnitude of force `vec|"F"|`) and effect (magnitude of displacement `vec|"r"|`) is a straight line passing through the second and fourth quadrant.
By measuring slope `1/"k"`, one can find the numerical value of force constant k.
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