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Kinematic equations of SHM in terms of displacement

The equations for the velocity $$(\vecphy{v})$$ and acceleration $$(\vecphy{a})$$ in terms of displacement $$(\vecphy{x})$$ can be derived from the equations in terms of equilibrium or extreme positions and the equation for displacement in terms of time ($$x=x_{0}\sin\omega t$$).

Like all kinematics equations, these can only be applied to linear motion with respect to a particular axis with a specified positive direction. This means that the quantities are represented as scalars, with negative signs to imply motion in the opposite direction.


The equation for acceleration is the defining equation for simple harmonic motion (showing the proportionality of $$a$$ and $$x$$).

Some remarks:

  • $$x$$ is and must always be smaller than $$x_{0}$$ for the velocity to be a real number.

  • The $$\pm$$ sign corresponds to the fact that the harmonic oscillator periodically changes direction, with velocities of the same magnitude but opposite directions occurring at the same displacement.

  • The minus sign in $$a=-\omega^{2}x$$ stands for the fact that the oscillator strives to restore the equilibrium whenever the displacement is non-zero.

$$x_{0}=$$maximum displacement; $$\omega=$$angular frequency.