Question

Horizontal Block-Spring mass system, x= 2.0m Sin (3.0t) t is in seconds. (a) find the period and frequency (b) find the speed, position and acceleration at t=2.4. (c) find the mechanical energy of the system, spring constant and amplitude. Thank you in advance.

Answer 1

a) Period, T = 2π/ω = 2π / 3.0 = 2.1 s

Frequency, f = ω/2π = 3.0 / 2π = 0.48 Hz

b) At t = 2.4 s, x = 2.0 * sin(3.0 * 2.4) = 1.6 m

v = dx/dt = (2 * 3) cos(3.0 t) = 6.0 cos(3.0 t)

At t = 0, v = 6.0 * cos(3.0 * 2.4) = 3.6 m/s

a = dv/dt = (6.0 * 3.0) * [-sin(3.0 t)] = -18 sin(3.0 t)

At t = 2.4 s, a = -18 * sin(3.0 * 2.4) = -14.3 m/s²

c) Amplitude, A = 2.0 m

ω² = k/m

=> Spring constant, k = mω² = m * (3.0)² = (9m) N/m

Mechanical energy of the system = PE_max = kA²/2 = (9m) * 2.0² / 2 = (18m) J