A 86 g particle undergoes SHM with an amplitude of 1.3 mm, a maximum acceleration of magnitude 6.8 x 103 m/s2, and an unknown phase constant φ. What are (a) the period of the motion, (b) the maximum speed of the particle, and (c) the total mechanical energy of the oscillator? What is the magnitude of the force on the particle when the particle is at (d) its maximum displacement and (e) half its maximum displacement?

mass, m = 86 g

Amplitude, A = 1.3 mm

Maximum acceleration, a = 6.8 x 10^3 m/s^2

(a) Let T be the period of motion and ω be the angular frequency.

maximum acceleration, a = ω²A

6.8 x 10^3 = ω² x 1.3 x 10^-3

ω = 2287.0875 rad/s

T = 2π/ω

T = (2 x 3.14) / 2287.0875

T = 2.75 x 10^-3 second

T = 2.75 milli second

(b) maximum speed = ωA

v = 2287.0875 x 1.3 x 10^-3

v = 2.97 m/s

(c) Total mechanical energy

T = 1/2 mω²A²

T = 0.5 x 86 x 10^-3 x 2287.0875 x 2287.0875 x 1.3 x 1.3 x 10^-6

T = 0.38 J

(d) At maximum displacement, the acceleration is maximum

F = m x a = 86 x 10^-3 x 6.8 x 1000

F = 541.8 N

(e) acceleration a = ω² y = ω² x A / 2

a = 2287.0875 x 2287.0875 x 1.3 x 10^-3 / 2

a = 3400 m/s^2

Force, F = 86 x 10^-3 x 3400

F = 292.4 N