Final answer:
The equation for the displacement of the mass in a mass-spring system is x(t) = 0.1 * cos(4t) + 0.25 * sin(4t). To plot x(t) and x'(t) for five periods of oscillations, you can use the equations x(t) and x'(t) and plug in values of t from 0 to 10π.
Step-by-step explanation:
The equation of motion for a mass-spring system can be given by the second-order differential equation:
m * x''(t) + k * x(t) = 0
where m is the mass, x(t) is the displacement of the mass, x''(t) is the acceleration of the mass, and k is the spring constant.
Given that the mass is 1 kg and the spring constant is 16 N/m, we can substitute these values into the equation to get:
1 * x''(t) + 16 * x(t) = 0
This equation can be rearranged to:
x''(t) + 16 * x(t) = 0
This is a second-order linear homogeneous ordinary differential equation with constant coefficients. The general solution to this equation is:
x(t) = A * cos(4t) + B * sin(4t)
where A and B are constants determined by the initial conditions.
Given that x(0) = 0.1 m and x'(0) = 1 m/s, we can substitute these values into the equation and its derivative to solve for A and B. The final equation for the displacement of the mass is:
x(t) = 0.1 * cos(4t) + 0.25 * sin(4t)
To plot x(t) and x'(t) for five periods of oscillations, we can use the equation and its derivative:
x'(t) = -0.4 * sin(4t) + 1 * cos(4t)
By plugging in values of t from 0 to 10π (five periods) into these equations, you can generate a set of points to plot the graphs.