THE MASS SPRING SYSTEM
OBJECTIVES
i. To examine the harmonic oscillations by modelling the motion of a mass-spring system with differential equations.
ii. To determine the effect of parameters on the solutions of differential equations.
iii. To determine the behavior of the mass-spring system from the graph of the solution.
iv. To determine the effect of the parameters on the behavior of the mass-spring.
1. Answer the following questions
a. Which curve represents y = y(t)? How do you know?
b. What is the period of the motion?
c. We say that the mass comes to rest if, after a certain time, the position of the mass remainswithin an arbitrary small distance from the equilibrium position. Will the mass ever come torest? Why
d. What is the amplitude of the oscillations for y?
e. What is the maximum velocity (in magnitude) attained by the mass, and when is it attained?
f. How does the size of the mass m and the stiffness k of the spring affect the motion?
2. The energy of the mass-spring system is given by the sum of the potential energy and kinetic energy.
a. Plot the quantity E as a function of time. What do you observe?
b. Show analytically that energy is constant.
c. Plot v versus y (phase plot). Does the curve ever get close to the origin? Why or why not? Whatdoes that mean for the mass-spring system?
3. Fill in LAB05ex1a.m to reproduce
a. |y|<0.02 for t>t1 with 5.581b. What is the maximum (in magnitude) velocity attained by the mass, and when is it attained?
c. How does the size of c affect the motion?
d. Determine analytically the smallest (critical) value of c such that no oscillation appears inthe solution.
4.
a. Plot the quantity E as a function of time
b. Show analytically
c. Plot v versus y (phase plot).