Q4. What is the phase difference between x1 = A sin ωt and x2 = A cos ωt? Which leads?
A sin ωt = A cos(ωt − π/2), so the phase difference is π/2, which is a time difference of T/4. The cosine motion leads: it reaches the corresponding position with the same velocity direction T/4 earlier, modulo a full period.
Q5. A particle in SHM has amplitude 5 cm and period 4 s. At t = 0 it is at the mean position, moving towards +x. Write x(t) in the form A cos(ωt + φ) and find x at t = 1 s and t = 3 s.
ω = 2π/4 = π/2 rad/s. Starting at the mean position moving towards +x means φ = −π/2, so x = 5 cos(πt/2 − π/2) cm, which is 5 sin(πt/2) cm. At t = 1 s, x = 5 cm; at t = 3 s, x = −5 cm.
Q6. A particle in SHM with period 1.2 s starts from rest at x = A. How long does it take to reach x = A/2?
x = A cos ωt, so A/2 is reached when cos ωt = ½, that is ωt = π/3. Then t = T/6 = 1.2/6 = 0.2 s.