Simple Harmonic Motion (SHM)
A simple harmonic motion (SHM) is described by a moving point over a circle (radius equal to the peak amplitude) with a constant angular velocity \((\omega)\).
The projection of the motion the point would appear to perform a to and fro motion perpendicular to the plane of the circle.
SHM is defined as the motion in which acceleration is proportional to the displacement and always directed towards the mean position.
The motion of a point over a circle can be seen by increasing the value of \(\alpha\) from \(0^\circ\) (saffron coloured slider) for the sine wave, which indicates that at time \(t=0\), displacement is zero. Hence, the SHM can be expressed as \(y=A\sin(\omega t)\).
For the cosine wave, movement of the point can be seen from \(\beta=0^\circ\,(\alpha=90^\circ)\), which indicates that at time \(t=0\), displacement is equal to the magnitude of the peak amplitude \((A)\) (can be varied by using the green coloured slider). Hence, the SHM can be expressed as \(y=A\cos(\omega t)=A\sin\bigg(\omega t+\dfrac{\pi}{2}\bigg)\).
Observation : The phase difference between the sine and cosine waves is \(\pi/2\).
The projection of the motion the point would appear to perform a to and fro motion perpendicular to the plane of the circle.
SHM is defined as the motion in which acceleration is proportional to the displacement and always directed towards the mean position.
The motion of a point over a circle can be seen by increasing the value of \(\alpha\) from \(0^\circ\) (saffron coloured slider) for the sine wave, which indicates that at time \(t=0\), displacement is zero. Hence, the SHM can be expressed as \(y=A\sin(\omega t)\).
For the cosine wave, movement of the point can be seen from \(\beta=0^\circ\,(\alpha=90^\circ)\), which indicates that at time \(t=0\), displacement is equal to the magnitude of the peak amplitude \((A)\) (can be varied by using the green coloured slider). Hence, the SHM can be expressed as \(y=A\cos(\omega t)=A\sin\bigg(\omega t+\dfrac{\pi}{2}\bigg)\).
Observation : The phase difference between the sine and cosine waves is \(\pi/2\).