The physics
A point turns on a circle of radius A at a steady rate ω. Its shadow on the x axis moves in simple harmonic motion (SHM).
x = A cos(ωt + θ₀) v = −Aω sin(ωt + θ₀) a = −ω²x
A block on a spring has a = −(k/m)x. So it moves like the shadow when ω² = k/m.
ω = √(k/m) T = 2π/ω v = ω√(A² − x²)
The start angle θ₀ fixes where the block starts and which way it moves. JEE often writes the sine form x = A sin(ωt + φ). That is the shadow on the y axis.
Time on the circle goes with the angle turned. So equal distances near an end take longer than near the centre. One whole swing covers a path of 4A.
Two SHMs with the same ω are two arrows turning together. Their sum is one arrow, the two added head to tail.
R² = A₁² + A₂² + 2A₁A₂ cos δ