How Do You Apply Initial Conditions to Simple Harmonic Motion?

To determine a simple harmonic motion trajectory, you need the oscillator’s angular frequency and its initial position and velocity. The initial position sets where the motion starts; the initial velocity sets which way it is moving and how much of the oscillation is in the sine component.

The most direct method is to use a sine-and-cosine form of the solution. This avoids ambiguity about the phase angle and works for any initial position and velocity.

Start from the equation of motion

For an ideal oscillator, displacement $x$ from equilibrium satisfies

$$\ddot{x}+\omega^2x=0,$$

where $\omega>0$ is the angular frequency, measured in radians per unit time. For a mass $m$ attached to a spring of stiffness $k$, $\omega=\sqrt{k/m}$.

A general solution is

$$x(t)=C\cos(\omega t)+D\sin(\omega t),$$

where $C$ and $D$ are constants. There are two constants because the equation is second order, so you need two independent initial conditions to specify one particular motion.

Apply the initial conditions

Suppose the initial conditions are given at $t=0$:

$$x(0)=x_0,\qquad \dot{x}(0)=v_0.$$

First evaluate the displacement at zero. Since $\cos(0)=1$ and $\sin(0)=0$,

$$x(0)=C=x_0.$$

Now differentiate to find velocity:

$$\dot{x}(t)=-C\omega\sin(\omega t)+D\omega\cos(\omega t).$$

At $t=0$, this becomes $\dot{x}(0)=D\omega=v_0$, so $D=v_0/\omega$. Therefore,

$$\boxed{x(t)=x_0\cos(\omega t)+\frac{v_0}{\omega}\sin(\omega t)}.$$

This form is often the easiest one to use: insert the measured or stated initial position and velocity directly.

Worked example

A particle oscillates with angular frequency $\omega=4\ \text{rad/s}$. At $t=0$, it is $0.030\ \text{m}$ to the right of equilibrium and moving left at $0.080\ \text{m/s}$. Thus $x_0=0.030\ \text{m}$ and $v_0=-0.080\ \text{m/s}$. Substituting gives

$$\begin{aligned} x(t)&=0.030\cos(4t)+\frac{-0.080}{4}\sin(4t)\\ &=0.030\cos(4t)-0.020\sin(4t)\quad\text{m}. \end{aligned}$$

The negative sine coefficient represents the initial motion to the left. Check the initial values:

$$x(0)=0.030\ \text{m},\qquad \dot{x}(0)=-4(0.020)=-0.080\ \text{m/s}.$$

The same trajectory can be written as a single cosine, $x(t)=A\cos(\omega t+\phi)$. Comparing coefficients, or using the initial conditions, gives

$$A=\sqrt{x_0^2+\left(\frac{v_0}{\omega}\right)^2} =\sqrt{(0.030)^2+(0.020)^2}\approx 0.0361\ \text{m}.$$

For this cosine-plus-phase convention, $\phi=\operatorname{atan2}(-v_0/\omega,x_0)$. Here $\phi\approx0.588\ \text{rad}$, so an equivalent expression is $x(t)=0.0361\cos(4t+0.588)\ \text{m}$. The function $\operatorname{atan2}(y,x)$ uses both inputs to select the correct quadrant; this matters when the initial position or velocity is negative.

If the initial time is not zero

If the conditions are specified at time $t=t_0$, use elapsed time $t-t_0$:

$$x(t)=x_0\cos\bigl(\omega(t-t_0)\bigr)+\frac{v_0}{\omega}\sin\bigl(\omega(t-t_0)\bigr),$$

where $x_0=x(t_0)$ and $v_0=\dot{x}(t_0)$. This ensures that substituting $t=t_0$ returns the stated position and velocity.

Common mistakes

• Using position alone: Knowing $x_0$ does not usually fix the motion. At the same position, the oscillator could be moving in either direction or momentarily at rest. You also need $v_0$.
• Forgetting the factor of $\omega$: The sine coefficient is $v_0/\omega$, not $v_0$. This is necessary because differentiating a sine term introduces a factor of $\omega$.
• Mixing phase conventions: For $A\cos(\omega t+\phi)$, the initial velocity is $-A\omega\sin\phi$. Changing the plus sign in the phase to a minus sign changes the phase relationship.
• Using frequency in hertz as angular frequency: If a frequency $f$ is given in cycles per second, convert it using $\omega=2\pi f$ before using these formulas.
• Confusing displacement with absolute position: In the oscillator equation, $x$ is measured from the equilibrium position. If the equilibrium is at a nonzero coordinate, shift coordinates before applying the formula.

The result assumes ideal, undamped simple harmonic motion with constant angular frequency. Damping, a driving force, or a nonlinear restoring force changes the governing equation and generally changes the solution.