Phase Intervals, Travel Time, and Distance
For x = A cos theta with theta advancing uniformly at omega, elapsed time is Delta theta/omega; total distance must be accumulated across every turning point crossed.
Why this shows up in the exam
Time between specified positions · Distance versus displacement over many cycles · First-arrival and repeated-phase questions
Learn the idea
Equal distances in SHM generally take unequal times because phase, not position, advances uniformly. The oscillator rushes through the centre and slows near each extreme, so time intervals are found by angles on the reference circle rather than distance ratios.
🧠 Memory hook: Convert positions to angles before comparing times.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- Delta t = Delta theta/omega — time corresponding to a directed phase change
- T = 2 pi/omega — one complete phase increase of 2 pi
- x/A = cos theta — position-to-phase conversion for a chosen origin
How to approach it
- 1Mark the starting phase and direction
- 2Trace the shortest allowed phase path
- 3Count complete cycles and turning-point segments separately
Common slip-ups that cost marks
- •Choosing the wrong phase branch
- •Ignoring direction of motion
- •Using net displacement as total distance
🌟 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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