Week 11 - Second Order Differential Equations

Question 1

a) Consider a mass of 1kg on a spring. It is described by the following differntial equation, x’’(t)= - (k/m)*x(t). Use Mathemtica to solve this.

b) Add in the initial conditions x(0)=0 and x’(0)=1 to the above differential equation and solve it.

c) Physically what do x(0)=0 and x’(0)=1 correspond to?

Text Answer:

d) Choose a value of ‘k’ and plot your solution. Is your plot consistent with the intial conditions and your answer to part ‘c’?

Text Answer:

Question 2

Prepare for the question by ensuring the variables we are going to use aren’t assigned to anything else by evaluating the cell below:

Remove[x,t];
Clear[m];

a) Now consider the same mass of 1kg but with a spring that is damped. It will be described by the following differential equation: x’’(t)= - b*x’(t)- (k/m)*x(t). Use Mathematica to solve this.

b) Add in the initial conditions x(0) =0 and x’(0) =1. Set k=10 kg/s^2

c) Make a function of ‘b’ and ‘t’ from your solution to the differential equation

d) Use your function to plot the evolution of the motion of the mass with a damping constant of b=1 over a range t=0 to 10s. (Hint: try using PlotRange-> All if some of your graph is not shown)

e)  Do you get a reasonable answer? In particular is your plot consitent with your initial conditions and does the plot show motion that shows some sort of damping?

Text answer:

f) Now try plotting your functino with a damping constant of b=10.