← all cardsTopic 8 · Oscillations · Equations & the two systems
Equations of motion — pick by the start
Starts at equilibrium: x = x₀ sin ωt. Starts at an extreme: x = x₀ cos ωt. Differentiate for v, again for a. Calculator in radians, always.
x = x₀ sin ωt → v = ωx₀ cos ωt → a = −ω²x₀ sin ωtWhat this actually means
The choice of sin or cos is set entirely by where the motion is at t = 0. Starting at equilibrium: sin (begins at zero, rising). Starting at an extreme: cos (begins at maximum).
Differentiate to descend the ladder from x to v to a. Each step multiplies the amplitude by ω and shifts the phase by a quarter cycle.
Radians, always. Degree mode silently wrecks every substituted value, and the error propagates through the whole question.
The trap
Choosing sin vs cos without checking the t = 0 position — every later part inherits the error.
Prove it — watch it be true
- Release the mass from an extreme and watch the x–t graph start at a peak: cos
- Imagine the clock starting as it crosses equilibrium instead: the same curve read as sin
- Check the v–t graph: same shape, shifted a quarter cycle, amplitude ωx₀