C.1.9 (HL)—SHM equations

HL motion equations

$x=x_0\sin(\omega t+\phi)$
$v=\omega x_0\cos(\omega t+\phi)$
$v=\pm\omega\sqrt{x_0^2-x^2}$

The sign of $v$ records direction; radians are used for phase.

HL energy equations

For an ideal oscillator,

$E_T=\frac12m\omega^2x_0^2,\qquad E_p=\frac12m\omega^2x^2,\qquad E_k=E_T-E_p$

At $|x|=x_0$ , $E_p=E_T$ ; at $x=0$ , $E_k=E_T$ .

Stable solution order

Convert all quantities to SI units, find $\omega=2\pi f$ , evaluate the phase in radians, substitute, and retain the velocity sign. Check that $|x|\le x_0$ and that each energy lies between zero and $E_T$ .

Common trap

Do not omit $\omega$ from the velocity equation, confuse $x$ with amplitude $x_0$ , or apply the quantitative energy equations to SL-only work.

How it is assessed

Questions ask for instantaneous velocity or maximum speed. The evidence rewards using the cosine velocity equation with the given phase and calculating ω before substitution; an energy method can be an accepted alternative for maximum speed when justified.

Command terms

Determine / Calculate

What earns marks

Convert f to ω = 2πf, use SI units, and substitute the same phase angle into x = x0 sin(ωt + φ) or v = ωx0 cos(ωt + φ). Show the sign of velocity and check vmax = ωx0.

Watch for

Using x0 sin(...) for velocity or omitting the factor ω in v = ωx0 cos(...).

Representative question

[Maximum number: 2]

Determine the vertical velocity of P at $t=3.0 \mathrm{~s}$ .

Use of $v=\omega x_{0} \cos (\omega t+\phi)$ and $\phi=\frac{\pi}{4}$ $v=\omega x_{0} \cos (\omega t+\phi)=(2.22)(0.9) \cos \left((2.22)(3)+\frac{\pi}{4}\right)=0.79 m s^{-1} \quad$

Set phase

Use radians and the phase angle to describe the initial condition: $x=x_0\sin(\omega t+\phi)$ . A phase difference of $\pi/2$ is a quarter-cycle offset.

Solve the equations

Use $v=\omega x_0\cos(\omega t+\phi)$ and $v_{\max}=\omega x_0$ . Convert $f$ to $\omega=2\pi f$ , use SI units, and retain the sign of velocity when direction is required.

Check the phase relationships

At an extreme, displacement is maximum and velocity is zero; at equilibrium, displacement is zero and speed is maximum. Displacement and velocity are one quarter-cycle out of phase.

Common trap

Do not omit $\omega$ , use degrees in a radian calculation, or treat phase angle as an amplitude.

C.1.9 (HL)—SHM equations

Solve SHM with Displacement and Velocity Equations

C.1.9 (HL) Exam Analysis

Assessment in practice

Retrieve the HL C.1 Simple Harmonic Motion Model

Question 1