Lab 5 - Jahbriel Gann - MAT 275 Lab
Table of Contents
EX 1 ................................................................................................................................. 1
A) ..................................................................................................................................... 1
B) ..................................................................................................................................... 1
C) ..................................................................................................................................... 1
D) ..................................................................................................................................... 1
E) ..................................................................................................................................... 2
F) ..................................................................................................................................... 2
EX 2 ................................................................................................................................. 4
A) ..................................................................................................................................... 4
B) ..................................................................................................................................... 5
c) ...................................................................................................................................... 5
EX 3 ................................................................................................................................. 7
A) ..................................................................................................................................... 7
B) ..................................................................................................................................... 7
C) ..................................................................................................................................... 7
D) ..................................................................................................................................... 9
EX4 .................................................................................................................................. 9
A) ..................................................................................................................................... 9
B) ................................................................................................................................... 10
C) ................................................................................................................................... 10
The Mass-Spring System
EX 1
A)
We know the blue curve represents y=y(t) because in the code for lab05ex1 code for the plot: (t,y, 'bo-'
represents y as blue.
B)
The period is about 4 by looking at the graph. Solving using 2pi/wo the period equals 4.19
C)
It will never come to rest due to there not being any friction or damping and in relation, no loss of energy.
This can be seen on the phase plot where no values go to zero.
D)
The amplitude is 0.8 which is the positive value of y(0)
1
Lab 5 - Jahbriel Gann - MAT 275 Lab
E)
t= 0.83, t=2.95, t=5.09, t=7.09, t=9.27, t=11.34, t=13.51 the magnitude of v is about 1.235 for each t the
mass is 0 meters away when maximum velocity is attained
F)
by increasing the mass, the period will be longer if the k value stays the same. When the k value increases,
the period shrinks.
LAB05ex1
LAB05ex1a
LAB05ex1b
2
0.8
0.6
0.4
0.2
1.5
Lab 5 - Jahbriel Gann - MAT 275 Lab
3
Lab 5 - Jahbriel Gann - MAT 275 Lab
EX 2
A)
According to the graph being a circle, the energy of the function does not change
LAB05ex1c
4
Lab 5 - Jahbriel Gann - MAT 275 Lab
B)
dE/dt = .5m2v'v + .5k2yy' v'=y''= =(k/m)*y
= mv'v+kyy'
= -k*y'*y + k*y*y' = 0 = dE/dt
c)
The curve never gets close to the orgin because there is no damping. This means there is no energy loss.
LAB05ex1c
5
3.062
3.0617
3.06
3.059
3.058
3.057
3.056
10 15
1.5
0.5
7
-0.8
0.6
-0.4
“0.2
0.2 0.4 0.6
0.8
Lab 5 - Jahbriel Gann - MAT 275 Lab
6
Lab 5 - Jahbriel Gann - MAT 275 Lab
EX 3
A)
t value is about 4.7 < t < 4.9
B)
t=1 amplitude= 0.797
C)
The graphs show that it reaches equilibrium faster when the c value is higher. The larger the value of c,
the lower number of ocillations.
LAB05ex2
7
c=12
Lab 5 - Jahbriel Gann - MAT 275 Lab
8
Lab 5 - Jahbriel Gann - MAT 275 Lab
D)
There needs to be no repeated root for no oscillations.
% = r^2+2pr+wo^2
% =r1=r2= -p +- (p^2-wo^2)^.5
% =p^2-wo^2 = 0
% =c^2-4m^2wo^2 = 0
% c= 6
EX4
A)
LAB05ex4
% as time increases, the value goes to zero. Energy also decreases
9
Lab 5 - Jahbriel Gann - MAT 275 Lab
B)
E = .5*m*v.^2+.5*k*y.^2
dE/dt = mv*v' + ky*y'
v' = y'' = -w0^2-2pv
w0 = sqrt(k/m) p = c/(2m)
= -k*y*y'-c*y'*y'+k-y-y'
= -c*y'*y'
dE/dt is greater than 0 when c is negative
dE/dt is less than 0 when c is positive
C)
The graph goes to 0,0 because in a spring system, the energy decreases
overtime.
10
