P2820 Final Exam W2013

Name:
Student number:

There are 4 questions for a total of 45 marks.  This is a 2-hour, open book exam.

Q1	[15 marks]

A Hooke’s Law spring has an equilibrium length L=0.50m and stiffness constant k=28.20 N/m.  It is hung vertically from one end as shown below (the top of the spring is at y=0).  To the other end is attached a ball of mass m=0.30 kg.  The ball is pulled downward, stretching the spring by an amount Δ=0.40m, and released from rest.  During its motion, which is only in the vertical direction, the mass feels a drag force of magnitude 0.246 v, where v is the speed of the mass.  Use g=9.81 m/s/s.

a) 6 marks

Solve for the position of the mass for the first 10s of its motion, using one of Mathematica’s built-in ODE solvers.  Plot your result.

Remove["Global`*"]

m=0.3;
k=28.20;
g=9.81;
b=0.246;

b) 1 mark

Using simple arguments for force balance, determine the position of the mass when it comes to rest.

c) 6 marks

Using the 4th-order Runge-Kutta algorithm from class, code your own numerical solution to part a.  Plot your solution.  Also plot the difference between your solution and the solution form part a.  Use a timestep of Δt=0.01.

d) 1 mark

How would you expect the plot of the difference between your solution and Mathematica’s to change if Δt were decreased by a factor of 10?  Why?  (You can also simply illustrate what happens and provide a rationale for the behaviour.)

e) 1 mark

Describe qualitatively what would happen to the motion of the mass if the drag coefficient b were to double.

Q2	[12 marks]

Download the file vcar.dat from the course website.  It contains two columns of data: time (s) in the first, speed (m/s) in the second.  The data represent the slowing down of a racecar after it deploys a braking parachute.

a) 2 marks

Import and plot the data.

b) 4 marks

Determine the accelation of the car as a function of time using the (discrete form of) the centred difference approximation for the first derivative of a function.  Plot the acceleration as a function of time.

c) 6 marks

The parachute provides a drag force of magnitude b (v^2), where v is the speed of the car.  Determine the drag coefficient b to 3 significant figures by appropriately plotting the data and determining the slope through a linear least-squares fit.  The mass of the car 1420kg.

Q3	[14 marks]

A particle of unit mass initially at the origin and with velocity of 0.6 m/s in the x-direction feels a force governed by a potential energy given by

U(x, y) = sin(3x) cos(2y)

a) 2 marks

Plot the potential energy surface for -2 ≤ x ≤2 and -2 ≤ y ≤2

b) 2 marks

Find expressions for the x and y components of the force on the particle due to the potential energy.

c) 6 marks

There is an additional drag force of -0.1 v, wher v is the velocity of the particle.  Use NDSolve to solve for the x and y coordinates of the particle as functions of time from t=0 to t=10s.  Plot the trajectory of the particle in the xy plane.

d) 4 marks

Plot the kinetic energy, potential energy and total (mechanical) energy of the particle for the first 10s.  Qualitatively, what characterizes the regions in time during which the particle loses mechanical energy at comparatively large rates and why does this make sense?

Q4	[4 marks]

A P2820 student attempts to solve the wave equation for a string of length 1m with ends fixed at zero displacement and the following initial conditions: the initial displacement is sin(3π x) exp(-x*x) in cm, with x being the position along the string in metres, and initial velocity is sin(4π x) in cm/s.  The (phase) speed of the wave on the string is 0.8 m/s.  The student wants to plot the profile of the string at t = 5 s.  Below is the student’s work.  Find the 4 errors to to make the correct plot.

Remove["Global`*"]

s[x_,t_]=w[x,t]/.NDSolve[{(∂_(t,t))w[x,t]==.64 (∂_(x,x))w[x,t],w[x,t]==0.01Sin[3π x] Exp[-(x^2)],Derivative[1,0][w][x,0]==0.01Sin[4π x],w[t,0]==0,w[t,1]==0},w[x,t],{t,0,5},{x,0,1}]//First

Plot[s[x,5],{x,0,1},AxesLabel->{"Position along string [m]","Displacement of string [m/s]"}]