In simple harmonic motion, motion is executed by a particle that is subject to a force which is ________ to the displacement of the particle and is directed toward the _________.
Simple harmonic motion
Mathematical view
Step 1: Velocity of the harmonic pendulum
Considering a simple harmonic wave
Where A is the amplitude, y is the displacement and is the angular velocity of the harmonic wave.
Differentiating y with respect to time, we get
Velocity,
Step 2: Acceleration of the harmonic pendulum
Again differentiate we get,
Acceleration, , (since ).
Or
or
The negative sign shows that the displacement and acceleration of the particle is directed to the opposite direction. i.e., the mean point
Step 3: Exerted force on the harmonic pendulum
From newtons law of motion, , where m and a are the mass and acceleration of the oscillator.
From equation ,
Force,
Force is directly proportional to displacement.
Therefore, it is clear that force is proportional to the displacement and acceleration (force) is directed towards the mean point.