Friction What have we learned from graphing our ideal velocity against our actual velocity from the ramp exercise We lose a lot of energy due to outside forces such as drag friction the coaster wobbling heat loss etc ID: 778124
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Slide1
Physics is PHUN!!!
http
://engineering.purdue.edu/Step/class_material
Slide2FrictionWhat have we learned from graphing our ideal velocity against our actual velocity from the ramp exercise?We lose a lot of energy due to outside forces, such as drag, friction, the coaster wobbling, heat loss, etc.To account for this, we’ve created equations of loss to use
for your modeling.
These equations were created by doing many trials and mathematical calculations and have been simplified as much as possible for your use.
Slide3Loss EquationsThe upcoming loss equations are set up as follows:Find the ideal velocity at the location, as you have been.Subtract the loss from the ideal velocity, and you will have the actual velocity.You must separate your
coaster’s track
into
sections where it is
decelerating, accelerating, and traveling on a
horizontal
surface.
In the equations,
∆x
is the distance from the start of your
section
(where your coaster first started decelerating or accelerating) to your current
location,
in feet
.
L
is the
velocity loss
of the previous section (from where it started to decelerate or accelerate to where your current section started)
Slide4Frictional LossTo find the frictional velocity loss on any section of your coaster, use the following equation:L
n
=
v
ideal,n
–
v
actual,n
where…
L
: frictional velocity loss on current track section
v
ideal
: ideal velocity, according to Energy Conservation
v
actual
: actual velocity, calculated with formulas in the next three slides
Note: The very first loss for your coaster,
L
0
, will be equal to 0.
Slide5Friction - Accelerating When your car is accelerating (i.e. moving downhill), use the following are loss equations:
If the angle
is greater than
45
º (from horizontal):
v
actual,
curr
=
v
ideal,
curr
– (∆x +
L
prev
)
If the angle
is less than or equal to
45
º (from horizontal):
v
actual,
curr
=
v
ideal,
curr
–
(0.75*∆
x +
L
prev
)
If
the length of the piece of track is less than 1.5
ft,
no matter the angle
:
v
actual,
curr
=
v
ideal,
curr
–
(0.3
+
L
prev
)
where…
∆x
: length of
current
section
L
: loss from
previous
section
Slide6Friction - Decelerating If your coaster is decelerating (i.e. moving uphill):If your angle is greater than 45º (from horizontal):
v
actual,
curr
=
v
ideal,
curr
– (-∆x +
L
prev
)
If your angle is less than or equal to
45
º (from horizontal):
v
actual,
curr
=
v
ideal,
curr
–
(-0.75
*∆x +
L
prev
)
where…
∆x
: length of
current
section
L
: loss from
previous
section
Slide7Friction - Horizontal When you coaster is moving on a horizontal surface, use the following equation for loss:
v
actual,
curr
=
v
ideal,
curr
– (1.32*∆x +
L
prev
)
where…
∆x
: length of
current
section
L
: loss from
previous
section
Slide8h
1
h
3
h
2
∆x
1
∆x
2
θ
1
θ
2
Velocity Loss Example
Knowns:
Unkowns
:
∆x
1
= 5
θ
1
= 75⁰ h
1,
h
3,
∆x2 = 2 θ2 = 15⁰ Ideal and Actual VelocitiesV0 = 0 for each section Lprevious for each section
Slide9h
1
h
2
∆x
1
θ
1
h
1
= ∆x
1
* sin(
θ
1
) = 5 * sin(75) = 4.83
Section 1 –
D
ownward Ramp
mgh
1
+ .5mV
0
2
= mgh
2
+ .5mV
2
ideal
__________
V
ideal
= √ (2*32*4.83)
V
ideal
= ______
ft
/sec
=
V
actual
=
V
ideal
– (∆x
1
+
Lprev )
Vactual = 17.58 – (5 + 0)Vactual = ______ ft/sec
L1 = Videal - Vactual
L1 = 17.58 – 12.58 L1 = ___ ft/sec
Slide10h
3
∆x
2
θ
2
Section 2 – Upward Ramp
h
2
h
3
= ∆x
2
* sin(
θ
2
) = 2 * sin(15) = .52
mgh
2
+ .5mV
0
2
= mgh
3
+ .5mV
2
ideal
V
0
= Vactual,1
_____________________
V
ideal
= √ (12.58
2
- (2 * 32 * .52)
V
ideal
= _____
ft
/sec
V
actual
= 11.18 – (-0.75*2 + 5)
V
actual
= ___
ft
/sec
L
2
= Videal - Vactual
L2 = 11.18 – 7.68
L2 = ____ ft/sec
V
actual
=
V
ideal
– (-0.75*∆x +
L
prev
)
L
prev
= L
1
Slide11OverallSo, in the example we modeled before, the sections would break down as follows:Ramp…Accelerating the entire time
Downward Curve…
Accelerating the entire time
Turn…
Flat surface
Loop…
Decelerating until
90 degrees (from horizontal!)
Accelerating after
90 degrees (from horizontal!)
IF
you
use
a banked
turn…
Decelerating for the first half of the turn
Accelerating for the second
half of the turn
Slide12Team Modeling Exercise(That means as a TEAM!)As a team, calculate the actual velocities at all of the locations in your Excel spreadsheet.
Slide13Actual Velocities
Slide14TimeThe time at any location is simply the distance traveled since the velocity was last calculated divided by the velocity at that point.As a team, take 5 minutes and find the elapsed time for every velocity on your Excel spreadsheet.
Slide15Time
Should have 1ft intervals (whole numbers) in here
Slide16Is this realistic?Since the loss equations we are using are specific to our materials, we have to transfer our numbers to the size of an actual coaster. This means that the scale factor becomes 1 in. = 1 ft., so from the previous exercises, 40 ft. = 1 ft.This scale factor will not be used to determine velocity, but it should be used to determine elapsed time in your cost workbook.
Slide17Model Your Own Coaster!Now you get to model your own coaster!!!Your coaster should have the actual velocity, g’s, running distance (1 ft intervals)
, and running time at every increment
.
You should measure the track distance at the beginning and end of every section (broken down into track pieces (i.e. loops, ramps, etc.) and accelerating or decelerating) and at one foot intervals between those increments. For loops or curves use simple angle divisions (45 or 90 degrees) and not one foot intervals
.
Your sheet should also have a graph of the g’s vs. running distance and your estimated thrill factor and a graph of the velocity vs. distance
.
You may include any other graphs that will make your spreadsheet easier to understand. Don’t forget to format everything!
Slide18Analyzing Your DataGetting weird numbers???Need to make any changes???What kind of changes???Have you documented the changes???