Question

The energy of the mass-spring system is given by the sum of the kinetic energy and the potential energy. In the absence of damping, the energy is conserved.

(a) Add commands to LAB05ex1 to compute and plot the quantity E=mv² + ky² as a function of time. What do you observe? (pay close attention to the y-axis scale and, if necessary, use ylim to get a better graph). Include at least one plot. Does the graph confirm the fact that the energy is conserved?

(b) Show analytically that = 0. (Note that this proves that the energy is constant). dt

(c) Add commands to LAB05exi to plot v vs y (phase plot). Include the plot. Does the curve ever get close to the origin? Why or why not? What does that mean for the mass-spring system?

Answer 1

The energy of a mass-spring system is conserved and can be represented by the quantity E=mv² + ky². This can be plotted as a function of time to observe the conservation of energy. Additionally, the derivative of E concerning time is zero, proving that the energy is constant. The v vs y (phase plot) shows a curved curve that never reaches the origin due to the constant amplitude in a mass-spring system.

Step-by-step explanation:

(a) The energy of a mass-spring system is the sum of the kinetic energy and the potential energy. In the absence of damping, the energy is conserved. To plot the quantity E=mv² + ky² as a function of time, you can use the following commands in LAB05ex1:

E = m * v**2 + k * y**2
plt.plot(t, E)
plt.xlabel('Time')
plt.ylabel('E')
plt.show()

When you plot E as a function of time, you will observe that the energy remains constant at all times, confirming that the energy is conserved.

(b) To show analytically that dE/dt = 0, you can differentiate the expression for E concerning time. Since m, v, k, and y are all constants, the derivative of E concerning time is zero.

(c) To plot v vs y (phase plot), you can add the following commands to LAB05ex1:

plt.plot(y, v)
plt.xlabel('y')
plt.ylabel('v')
plt.show()

The curve in the v vs y plot does not get close to the origin because the mass-spring system is in simple harmonic motion, where the amplitude remains constant. This means that the mass-spring system never reaches the origin, resulting in a curved phase plot.