Question

An air-track glider attached to a spring oscillates with a period of 1.5 s. At t = 0 s the glider is 5.00 cm left of the equilibrium postion and moving to the right at 36.3 cm/s. a) What is the phase constant? b) What is the phase at t = 0 s, 0.5 s, 1.0 s, and 1.5 s?

Answer 1

Let's assume x(t) = Asin(ωt + φ)
Then v(t) = Aωcos(ωt + φ)

x(0) = 5 cm = Asin(ω*0 + φ)
v(0) = 36.3 cm/s = Aωcos(ω*0 + φ)

v(0) / x(0) = 36.3cm/s / 5cm = 7.26 rad/s = Aωcosφ / Asinφ = ω / tanφ
But ω = 2π/T = 2π / 1.5s ≈ 4.2 rad/s
tanφ = 4.2rad/s / 7.26rad/s = 0.577
φ = 30º = π/6

b) Not sure what you mean by phase. If you mean "the expression inside the trig function", then
phase(0) = π/6
phase (0.5s) = ωt + π/6 = 2π(0.5s)/1.5s + π/6 = 2π/3 + π/6 = 5π/6
phase (1.0s) = 2π(1/1.5) + π/6 = 8π/6 + π/6 = 3π/2
phase (1.5s) = 2π + π/6 = 13π/6 = π/6