Oscillation about a non-zero centre
● VERBATIM — examined word-for-wordTides, mass below a ceiling, floating tubes: the reading oscillates about a mean, not zero. Centre = (max + min)/2, amplitude = (max − min)/2, equation = mean + x₀ sin ωt. Trap: in a distance-from-ceiling graph the amplitude is NOT the max reading — subtract the centre first. Time between high and low = T/2.
reading = centre + x₀ sin ωtWhat this actually means
Tides, a mass hanging below a ceiling, floating tubes: the measured reading oscillates about a mean value, not about zero. First job: extract centre = (max + min)/2 and amplitude = (max − min)/2.
The amplitude is NOT the maximum reading. In a distance-from-ceiling graph, subtract the centre before calling anything an amplitude. Time from a high to the next low is T/2, not T.
Once centre and amplitude are extracted, everything is ordinary SHM about that centre: reading = centre + x₀ sin ωt, and all the standard formulas apply to the x measured from the centre.
Reading the amplitude as the maximum value on a distance-from-ceiling graph — subtract the centre first.
Prove it — watch it be true
- Note the x–t graph oscillates about zero only because x is measured from equilibrium
- A raw ruler reading (mass to ceiling) would trace the same curve lifted to a non-zero mean
- Recover the physics: centre = (max + min)/2, amplitude = (max − min)/2