**Example 1:** At time t = 0, the displacement is x = 2.72 m, the velocity is v = —2.54 m/s, and the acceleration is a = —10.87 m/s^{2}.

a) Find the amplitude, phase, and angular frequency for the harmonic motion.

b) Find expressions for x(t), v(t), and a(t).

**Solution**: We find for the ratio of the acceleration and coordinate,

**a/x = -(10.87 m/s ^{2}) / 2.72 m = – 4.0 s^{-2} = -ω^{2} → ω = 2.0 s^{-1}**

and the period is **T = 27π /w = 2π/(2.0s ^{-1}) = 3.1 s.**

**Example 2**: Find the period of a mathematical pendulum that is an object with mass in attached to a string of length L.

**Solution:** The force on the object is the weight F = mg (directed downward). For the torque about the fix point on the ‘ceiling: we get

**τ = -mg • L sin ϕ. **

Note that the torque is negative (clockwise) [positive (counter-clockwise)] when the angular displacement is positive, ϕ > 0 [negative ϕ < 0]. Since the moment of inertia is I = mL^{2}, we have τ = lα, so that **-mgL sin α = mL ^{2} sin ϕ , or α = (g/L) ϕ**

Then we find yields: **T = 2π √(L/g)**

**Discussion:** We find the length of a mathematical pendulum with period T = 2.0 s. We get **L = (9.8 m/s ^{2})/ (πs)^{2} = 1.0m.**