What is the maximum spring energy?

Question

A point body weighing 10 grams attached to a spring is a harmonic with an amplitude of 3 millimeters

makes a vibrating motion. The vibration period is 0.1 second. What is the maximum spring energy?

Answer 1

Let

m = 10 g = 0.01 kg be the mass attached to the spring,

A = 3 mm = 0.003 m be the amplitude,

T = 0.1 s be the period,

k (unknown) be the spring constant.

The spring energy is given by the formula

E = kA²/2

We do not know k, but we can calculate it from the given period, whose formula is

T = 2π√(m/k)

From that we can express

k = 4π²m/T²

Now we can calculate the energy

E = kA²/2 = 4π²mA²/2T² = 2m(πA/T)²

Substituting actual numbers

E = 2 x 0.01 kg (π x 0.003 m / 0.1 s)² = 1.78 x 10^-4 J