It is impossible for two oscillators, each excuting simple harmonic motion, to remain in phase with each other if they have different

In simple harmonic motion (SHM), the phase of an oscillator at any time 't' is given by the expression (ωt + φ), where ω is the angular frequency and φ is the phase constant. For two oscillators to remain in phase over time, their phase difference must remain constant. If two oscillators have different time periods (T), their angular frequencies (ω = 2π/T) will also differ. Consequently, the term ωt will change at different rates for each oscillator, causing their phase difference to increase or decrease continuously over time rather than remaining constant. While factors like amplitude, spring constants, and kinetic energy affect the magnitude or energy of the oscillation, they do not inherently prevent two oscillators from staying in phase as long as their frequencies (and thus time periods) are identical. Therefore, identical time periods are a necessary condition for maintaining a constant phase relationship.

Sources

1. [1] Science-Class VII . NCERT(Revised ed 2025) > Chapter 8: Measurement of Time and Motion > 8.1.1 A simple pendulum > p. 110