Lecture10_Springs
Download
Report
Transcript Lecture10_Springs
Recent exam scores are shown as a histogram
ordered simply by student ID number.
Exam Scores
100
90
80
70
60
50
40
30
20
10
0
Student ID number
What is the best approximation of
the class average on this exam?
A. 80
B. 60
C. 40
D. 20
The same scores are rearranged
in ascending order below.
Exam Scores
100
90
80
70
60
50
40
30
20
10
0
What is the best approximation of
the class average on this exam?
100
90
80
70
60
50
40
30
20
10
0
avg = 85
A. 80
C. 40
B. 60
D. 20
velocity, v
velocity, v
A.
B.
time (seconds)
velocity, v
velocity, v
time (seconds)
C.
time (seconds)
D.
time (seconds)
Which of the graphs above could
represent an object freely falling from rest?
velocity, v
velocity, v
vmax
1
2
vmax
For the moving object
graphed at left
time (seconds)
A. vavg < 12 vmax
vmax
B. vavg = 12 vmax
vmin
C. vavg > 12 vmax
velocity, v
velocity, v
time (seconds)
vmax
1 v
2 max
For the moving object
graphed at left
time (seconds)
A. vavg < 12 vmax
vmax
B. vavg = 12 vmax
C. vavg > 12 vmax
time (seconds)
Weight
Support (floor)
Adding all these supporting forces together
some pull left,
some pull right,
some pull forward,
some pull back
(a tug-of-war
balancing)
all tend to
pull UP!
Styrofoam bridge
weighted at center
Pressure applied to rigid glass bar
x
x
x
Natural length
x
W
F
Springs may be compressed (shortening their
“natural length”) until the “restoring force” it
counters with in an attempt to regain its original
shape exactly balances the applied force.
x
Springs may be elongated (beyond their
“natural length”) until the “restoring force”
exactly balances the force pulling it open.
Note:
F x
Compound bows
use systems of
pulleys and cams
to maintain a
fairly constant
resistance force
over the full
distance drawn.
Otherwise, for a more
conventional recurve
or simple longbow
Force
distance
to hold bow is
in place
drawn
Force
F x
Displacement, x
Also means:
F = kx
“spring constant” in units of N/m
The restoring force grows so many Newtons
for every meter stretched.
The weight of a 1250 kg car is evenly
distributed over 4 coil springs with strength
k = 32000 N/m.
When carrying 5 passengers (each, on
average 73.0 kg) how much lower does the
car ride?
The empty car already compresses each spring:
x F / k
¼th total weight, empty car
2
(1 / 4)(1250kg )( 9.8m / s )
32000 N / m
= 0.096 m = 9.6 cm
With everyone on board:
(1 / 4)(1250kg 365kg )(9.8m / s )
x
32000 N / m
= 0.124 m = 12.4 cm
2
It rides about 2.8 cm (a little more than 1 inch)
lower.
snapped garage
door spring
sprung bicycle
seat spring
cracked
suspension
coil spring
permanetly
deformed
Slinky!
plastic
deformation
elastic range
Force
failure
proportionality
limit
Displacement, x
Work done against an elastic force
ending with
force = kx
Force
W = Favg d
=
starting with
force 0
=
Fmax
x
2
1
k(x)2
2
Displacement, x
F
d
The “full draw weight’ of this bow is 20 pounds (90
Newtons) at the 24 inches (0.60 meter) that this archer
has drawn it. What is its effective spring constant?
k F / x 90N / 0.60m 1.5N / m
How much work was done in drawing the bow back?
W k ( x ) (1.5N / m)(0.60m)
0.27 N m
1
2
2
1
2
2
If all the energy goes into the 400 grain (26 gram)
arrow’s kinetic energy, what speed will it leave the
bow with? 0.27 Nm 1 mv 2 1 (0.026 kg )v 2
2
v 20.77m / sec
2
2
2
2
v 4.56m / sec
Total
Weight
A tennis ball rebounds straight up from
the ground with speed 4.8 m/sec.
How high will it climb?
It will climb until its
speed drops to zero!
v v0 at
g = -9.8 m/sec2
final speed = 0
t - (4.8m / s) ( -9.8m / s )
0.490 sec
2
and in that much time
it will rise a distance
2
d v0t 12 at
d (4.8m / s )(0.490s )
-
2
2
1
(9.8m / s )(0.490s )
2
d 1.176 m
or
since time up = time down
the distance it falls
from rest in 0.49 sec:
d at
1
2
2
d 12 (9.8m / s 2 )(0.490s )2
d 1.176 m
A tennis ball rebounds straight up from
the ground with speed 4.8 m/sec.
How high will it climb?
It will climb until all its kinetic energy has
transformed into gravitational potential energy!
1
2
mv 2
= mgh
2
v
h
2g
2
(4.8m / s )
h
2
2(9.8m / s )
= 1.176 meter
h
A skier starts from rest down a slope with
a 33.33% grade (it drops
a foot for every 3
F
horizontally).
W
3h
The icy surface is nearly frictionless. How fast is he
traveling by the bottom of the 15-meter horizontal drop?
The accelerating force down along the
slope is in the same proportion to the
weight W as the horizontal drop h is to the hypotenuse he rides:
The long way:
F
h
h
2
2
W
h 10
9h h
1
F
W
a
10
1
10
1
g
10
h 9h 47.434 meters long
2
2
So it takes him d 1 at
t
2d / a
2
The slope is
2
2
t 2( 10h) /( g / 10 ) 20h / g 5.533 sec
During which he builds to a final speed:
v v0 at 0 (
g
10
)(
20 h
g
) 17.146 m / sec
h
A skier starts from rest down a slope with
a 33.33% grade (it drops
a foot for every 3
F
horizontally).
W
3h
The icy surface is nearly frictionless. How fast is he
traveling by the bottom of the 15-meter horizontal drop?
The gravitational potential energy he
has at the top of the slope will convert
entirely to kinetic energy by the time he reaches the bottom.
The short way:
mgh
v
v
2
1
2
mv
2
2 gh
2
2(9.8m / s )15m
= 17.146 m/sec
D
C 2h
B
A
h
1.5h
h
½h
3h
3h
The bottom of the run faces a slope
with twice the grade (steepness).
Neglecting friction, the skier has just
enough energy to coast how high?
We know rubber tires are easily deformed
by the enormous weight of the car they
support…but not permanently. They
regain their round shape when removed.
This “spring-like”
resiliency explains
the rebound of all
sorts of balls.
Racquetball
rebounding
from
concrete.
Tennis ball
rebounding
from
concrete.
Airtrack bumper carts:
Notice that if the spring bumbers
reverberate (ring) this would have
to represent some energy that did
not get returned to forward motion!
Unlike the stored potential of the compressed
bumpers, this is not a “potential” energy
that can ever be recovered as kinetic energy.
This represents a fractional
loss in kinetic energy!
A purely ELASTIC COLLISION is defined
as one which conserve kinetic energy.
Look how a
racquetball
still undulates
after leaving
the floor!
These
vibrations
are a
wasted
form of
energy!