Posted: November 17th, 2015

# Numerical Modeling Extra Credit Project

Numerical Modeling Extra Credit Project

Describing the motion of a particle – its position, velocity and acceleration as a function
of time – can be achieved fairly easily if the system is not too complex. However, nature
quickly can become very complex, so much that an analytical solution (an equation for
position or velocity as a function of time) can be too mathematically complex to attain.
Numerical modeling techniques can often be used to solve such systems.
Most of these numerical modeling methods involve taking a rather complicated system –
perhaps one in which the acceleration varies with position, speed or time – and breaking
into small intervals (of distance or time) and assuming that the acceleration and /or
velocity are constant over each small interval. The smaller the interval, the closer the
approximation is to reality.
The Euler Method is a numerical modeling method that breaks a complicated motion into
discrete intervals of time, and assumes that the acceleration and velocity are constant over
each interval.
The relation between the acceleration and velocities for some small time interval ?t are as
follows:
v v a t
v t t v t a t t
t
v t t v t
t
v
a t
n ? n ? n ?
? ? ? ? ?
?
? ? ?
?
?
?
?
?1 ?1
( ) ( ) ( )
( ) ( )
( )
(This equation is exact if the acceleration is constant.)
We can similarly approximate the position after the time interval ?t as well.
x x v t
x t t x t v t t
t
x t t x t
t
x
v t
n ? n ? n ?
? ? ? ? ?
?
? ? ?
?
?
?
?
?1 ?1
( ) ( ) ( )
( ) ( )
( )
The familiar
2
2
1 a(?t)
term is missing because of the assumption that ?t is small. If ?t is
small, (?t)2
is very small.
The position and velocity of the object as a function of time can then be approximated by
a series of calculations as follows:
Step Time Position Velocity Acceleration
0 to xo vo
m
F t
a
o
o
?
?
( )
1
t t t
1 ?
o ? ? x x v t
1 ?
o ? o? v v a t
1 ?
o ? o?
m
F t
a ?
?
( )
1
1
2
t ? t ? ?t
2 1
x ? x ? v ?t
2 1 1
v ? v ? a ?t
2 1 1
m
F t
a ?
?
( )
2
2
.
.
.
n
.
.
.
t t t
n
?
n?1 ? ?
.
.
.
x x v t
n
?
n?1 ? n?1?
.
.
.
v v a t
n
?
n?1 ? n?1?
.
.
.
m
F t
a
n
n
?
?
( )
The Assignment
You may work in groups of up to three students if you desire. Turn in one assignment
per group.
a) (5 pts) Consider a mass sliding down a frictionless curve in the shape of a quarter
circle of radius 2.00 m as in the diagram. Assuming it starts from rest, use Euler’s
method to approximate both the time it takes to reach the bottom of the curve and its
speed at the bottom. You may either use a spreadsheet like MS Excel or you may
write and execute a computer program in the language of your choice. Do three
trials: ?t = 0.2s, ?t = 0.02s, and ?t = 0.002 s. Compare the predicted speed at the
bottom for each case to the accepted value of 6.261 m/s. (Calculate a percent error.)
Does the approximation improve as ?t becomes smaller?
b) (5 pts) Repeat (a), but this time assume a constant kinetic friction coefficient of ?k =
0.200. Again determine the time to the bottom and the speed at the bottom. . You
need only run one trial: ?t = 0.002 s. As you do not have a “correct” value to
compare, do not calculate a percent error.
m
v = ?
2 m
m
?
For both (a) and (b), turn in the spreadsheet file or the code and via the D2L dropbox.
If you write a program, you’ll need to bring a working computer with the program to
my office and demonstrate it. Include in a summary of your results.
Hints:
(a) Define the position and acceleration of the mass in terms of the angular position
?. Be careful – some computer/spreadsheet functions assume radians. (b) Because it
is moving along a circular track, the mass will experience a centripetal acceleration.
This will affect the normal force

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