a) ![x(t)=2.0 sin (10 t) [m]](https://tex.z-dn.net/?f=%20x%28t%29%3D2.0%20sin%20%2810%20t%29%20%5Bm%5D)
The equation which gives the position of a simple harmonic oscillator is:

where
A is the amplitude
is the angular frequency, with k being the spring constant and m the mass
t is the time
Let's start by calculating the angular frequency:

The amplitude, A, can be found from the maximum velocity of the spring:

So, the equation of motion is
![x(t)= 2.0 sin (10 t) [m]](https://tex.z-dn.net/?f=x%28t%29%3D%202.0%20sin%20%2810%20t%29%20%5Bm%5D)
b) t=0.10 s, t=0.52 s
The potential energy is given by:

While the kinetic energy is given by:

The velocity as a function of time t is:

The problem asks as the time t at which U=3K, so we have:

However,
, so we have

with two solutions:


c) 3 seconds.
When x=0, the equation of motion is:

so, t=0.
When x=1.00 m, the equation of motion is:

So, the time needed is 3 seconds.
d) 0.097 m
The period of the oscillator in this problem is:

The period of a pendulum is:

where L is the length of the pendulum. By using T=0.628 s, we find
