Respuesta :
Answer:
t = T/4
Explanation:
The power delivered to the mass by the spring is work done by the spring per second.
[tex]P = \frac{dW}{dt}[/tex]
The work done by the spring is equal to the elastic potential energy stored in the spring.
[tex]U = \frac{1}{2}kx^2[/tex]
The maximum energy stored in the spring is at the amplitude of the oscillation.
[tex]U_{max} =\frac{1}{2}kA^2[/tex]
So the first time the mass reaches to its amplitude can be found by the following equation of motion:
[tex]x = A\cos(\omega t + \phi)\\\phi = \pi/2 ~because ~at ~t= 0, ~ x = 0\\0 = A\cos(0 + \pi/2)\\x = A\cos(\omega t + \pi/2)[/tex]
When the mass reaches the amplitude:
[tex]A = A\cos(\omega t + \pi/2)\\1 = \cos(\omega t + \pi/2)\\\omega t + \pi/2 = \pi[/tex]
because cos(π) = 1.
[tex]\omega t = \pi/2[/tex]
Using ω = 2π/T,
[tex]\omega t = \pi/2\\\frac{2\pi}{T}t = \pi/2\\t = \frac{T}{4}[/tex]
