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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.

$$$\begin{align*}v&=\pm\omega\sqrt{x_{0}^{2}-x^{2}}\\a&=-\omega^{2}x\end{align*}$$$

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.