Sinusoidal State and Initial Phase
An ideal SHM may be written x = A sin(omega t + phi), where A is the nonnegative amplitude and phi is fixed by the simultaneous initial displacement and velocity.
Why this shows up in the exam
Finding phase constants · Recovering motion from initial conditions · Rewriting squared trigonometric forms as SHM
Learn the idea
Amplitude and phase encode the complete SHM state once angular frequency is known. A sine or cosine wave is a clock for the oscillator; the phase says where the particle is in its cycle and which way it is moving.
🧠 Memory hook: Position chooses the sine; velocity chooses the quadrant.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- x = A sin(omega t + phi) — general sinusoidal displacement
- v = A omega cos(omega t + phi) — velocity fixes the direction and phase quadrant
- A² = x0² + (v0/omega)² — amplitude from an initial state
How to approach it
- 1Write x0 and v0 at the same instant
- 2Solve amplitude before phase
- 3Check the phase quadrant with the direction of motion
Common slip-ups that cost marks
- •Using inverse sine without checking velocity sign
- •Allowing a negative amplitude instead of shifting phase
- •Reading omega as ordinary frequency
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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