The given displacement at time, t, is
p(t) = 2 sin(tĀ³) + 5 cos(tĀ³)
The initial equilibrium position is
p(0) = 5
To determine future equilibrium postions, define
f(t) = p(t) - 5 = 2 sin(tĀ³) + 5 cos(tĀ³ - 5
The derivative of f(t) is
f'(t) = (3tĀ²)[2 cos(tĀ³) - 5sin(tĀ³)]
Equilibrium is established when f(t) = 0.
To solve this equation numerically, we shall use the Newton-Raphson method, givenĀ by.
t(n+1) = t(n) - f[t(n)]/f'[(t(n)], n=0,1,2, ...,
As a guess, let (0) = 1.
The iterativeĀ solution for t is shown below.
Ā n Ā Ā Ā Ā t(n)
Ā --- Ā -----------
Ā Ā 0 Ā 1.0000
Ā Ā 1 Ā Ā 0.9344
Ā Ā 2 Ā 0.9147
Ā Ā 3 Ā 0.9130
Ā Ā 4 Ā 0.9130
The solution converges rapidly to t = 0.913 s.
The graphical solution (shown below) confirms the numerical solution.
Answer:
The weight first reaches the equilibrium position in 0.913 sec.
