ENERGY
Modeling Unit VII
Whiteboard the Lab Results:

Sketch the graph for your spring




Write the equation for your spring



Identify the color of your spring
Just the straight line (no points)
Label each axis with their proper units
Replace y and x with the actual variables from the graph
Include units for the slope and y-intercept
Write down what you think each of the following represent



slope
the y –intercept represent about the spring.
The Area between your line and the displacement (stretch) axis
Warm-up Question
The equation for a spring is:
FT = (25N/m) Δd + 2.5N
Determine the following:
a)The tension force (FT ) required to stretch
the spring 0.50 m.
b) The amount of mass you would need to
hang from the spring to stretch it 0.50m
Spring-Force Lab Results
50
45
40
F (N)
35
30
25
20
15
10
5
0
0
0.5
1
1.4
Δd (m)
How much mass needs to be added to the red spring to
stretch it approximately 0.50 m?
50
45
40
35
30
25
20
F (N)
F = (25N/m) Δd + 2.5N
15
10
5
0
0
0.5
1
F = (25 N/m) (0.50 m) + 2.5N = 12.5 N +2.5N = 15.0 N
FT = 15.0 N = Fg = mass (10N/kg)
Δd (m)
mass = 0.150 kg =150 g
Hooke’s Law for Springs
General Equation:
F  25N / md  2.5N


F = kDd + F0
“k” = spring constant (strength of spring)
Ideal springs have no preload (F0).
 Any
amount of force on a spring causes it to stretch.
F
T
 kd
Effort to Stretch Spring

The energy to stretch a spring involves the
combination of…
1.
2.

The force of tension in the spring (FT)
The displacement or stretch of the spring (Δx)
How do we represent this force-stretch
combination? … or the combination of two
values in general?
Review: finding displacement (Δd)

If you are moving your displacement depends on 2
things…
1.
2.

How fast you are traveling
How much time you are traveling at that speed
We found this graphically by finding the area under the
curve of a velocity versus time graph.
During Constant Velocity:
Δd = (2m/s)•6s = 12m
During Acceleration:
Δd = (½)•bh
= 9m
Representing Effort to stretch a spring:
Effort to stretch spring = area under FT vs. stretch (Δd) graph
50
45
40
FT
(N)
Effort =
35
30
25
20
15
10
5
0
0
0.5
1
Δd (m)
Finding Effort Mathematically


The effort exerted to stretch a spring can be
used to do things (launch a projectile, etc…)
We can say that ENERGY is stored in a
stretched or compressed spring.
Energy stored
In a Spring (Eel)
=
Area under
The graph
since FT  kd ...
1
Energy  kd d 
2
1
1
 bh  FT d
2
2
1
2
Eel  k d 
2
Energy Stored in a Spring (Eel)
1
2
Eel  k d 
2
N 2
Units :  m  N  m
m
Eel = Elastic Energy
k = Spring Constant
Δd = displacement or stretch
1 N  m  1J
Joules (J) are the standard units for ENERGY
 A calorie is also a unit of energy.
Money Analogy for Energy


Use a whiteboard to illustrate how money ($) is
used and moves in our society:
Consider the following questions?
 How
is $ moved or is transferred in society?
 What can you do with $?
 Where is $ stored when it is not being transferred?
 How do individuals get $?
 Etc... Be creative! (school appropriate and legal)
Important Ideas about Money
$ is transferred between people
 You get $ by working
 $ gives you the ability to do things
 $ is stored in different places (banks, pockets,
etc…)
 The flow of money can be cyclic

Analogy for Energy

Spring – our system, something that can store energy.
A

spring is like a bank, a place where $ can be stored.
Energy – gives you the ability to do things / change
things.
 Energy
is like $, which gives you the ability to do things in
society.
 It cannot be created or destroyed, just transferred from one
place to another.

“Working” – transferring energy from one system to
another.
 Working
= transferring $ from one person to another(losing
and gaining $)
Energy ($) Storage Accounts




Elastic Energy Account (Eel)– you can store energy in an
elastic material by working on it (apply a force over some
displacement)
Gravitational Energy Account (Eg) – you can store energy
in the Earth’s gravitational field by increasing the distance
between the earth and an object.
Kinetic Energy Account (Ek) – the energy stored in moving
objects.
Internal Energy Account (Eint) – the energy stored in the
random motion of atoms in a system. Measured by change
in temperatures.
Energy Bar Charts (LOL)

A visual way to account for the energy in a system. Where the
energy is being stored and any transfers, into or out of the
system (bank)
Worksheet 3a (Problem #1) v1

Final
Initial

Define the system as…
Spring + Cart + Earth
NO FRICTION
Spring
+ Cart
+ Earth
Worksheet 3a (Problem #1) v2

Final
Initial

Define the system as…
Spring + Cart + Earth
WITH FRICTION
Spring
+ Cart
+ Earth
Worksheet 3a (Problem #1) v3
Final
Initial


Define the system as…
Cart + Earth
NO FRICTION
Cart
+
Earth
+ work is done
on the cart by
the spring
Worksheet 3a (Problem #1) v4
Final
Initial


Spring
-Work is done
on the spring by
the cart
Define the system as…
the Spring
NO FRICTION
Kinetic Energy Account:

What visually tells us that an object has more or less
kinetic energy (Ek)?
 Its

velocity or speed
…so the kinetic energy an object has depends on its
velocity.
J  N m
Studying Kinetic Energy (Ek & V)


N  kg 
m
s
s
If we pull the cart back, what kind of energy is stored in
the system?
Where does the energy go when you release the car and
the spring has lost all its stored energy?
Ek(J) V(m/s)
1
2
Eel  k x   Ek
2
Motion Sensor
Ek
?
(J)
0
Δx
m = 0.291kg
0
V(m/s)




The energy stored in moving objects.
Depends on the mass and velocity of an object.
Kinetic energy increases as mass increases.
Kinetic energy increases when velocity increases.
1
Ek  mV
2
2
Energy Review

What is energy?
 The
ability to do something
 Gained or lost through “working” (W=FΔx)

How or where can energy be stored?
 In
a stretched spring or elastic material
(Eel = 1/2k(Δx)2)
 In a moving object (Ek = 1/2mV2)
 By raising an object off the ground (Eg =?)
 In the motion of atoms or molecules (Eint)
Gravitational Energy (Eg) Account

To be above the ground
something has to do
work ON the car to give
it some Eg:
F =_____
h
Eg =
Fg = mg
g = 10N/kg
Whiteboard Problem

How high should the cart be placed so that it will
have a velocity of 1m/s when it goes through the
photogate?
(Photogate)
m = 0.291kg
h=?
h = 0m
Solving Other Problems

Changing gravitational energy to kinetic energy is
useful for solving many different types of problems.
Pendulums
1
Straight
Ramps
Freefall
Curved
Ramps
1
1
1
h
2
2
2
2
The speeds will be the same, but the directions different!
WK3a Problem #2
Final
20m
Initial

What is the car’s velocity
halfway up the loop?
Cart
Spring
Earth
m = 500kg
k = 8000N/m
Δx = 5m
h = 0m
h = 10m
V=?
WK3a Problem #8

How much force did
Super Man use when
stopping the train?
Train
Superman
Initial:
m = 100,000kg
V = 22.7m/s
or 50mi/hr
Final:
Δx = 50m
V = 0m/s
F=?
Whiteboard Problem

You are driving along at 22.7m/s (50mph)
on a wet country road late at night when a
deer jumps out 20m in front of your car. If
you immediately slam on your breaks how
fast will you be going when you hit the deer?
The car’s mass is 1500kg and the coefficient
of kinetic friction is 0.6
WK3b Answers #1 & 2
1. Δx = 2m
100J
100J
Cart
+ Earth
+ Spring
2. V = 9.5m/s
v=0
5
height (m)
1000J
m = 20 kg
v=?
Cart
+
Earth
900J
0
100J
WK3b Answers #3 & 4
3. h = 0.9m
m = 500 g
v=0
4.5J
Block
+ Earth
+ Spring
1,531J
Bullet
+ Earth
k = 100
x = 0.30 m
Initial
4.5J
0
Final
4. Δx = 0.03m
?
Initial
Final
W = -1,531J
?
Unit VII WK3b Answers

5. a. discuss
b.
c. Working by engine = work done by friction = 1000J
d. Since the Fengine> fk then the box will accelerate: a =
0.5m/s2

6. Δx = 20m

7. b. Fx = 86.6N  W = 866J
c. the box is accelerating because Fx > fk
d. Working by friction = 675J
e. discuss
Δx=0.35m
Unit VII WK4
Answers

F=?
1. a.
Ek = W
ball
-W done
by glove on ball
b. F = 180N

2. a. Eg = 6000J
c. V = 14.14m/s
b. Ek = Eg = 6000J
d. V = 20m/s or 1.4 times more
e. htotal = 40m or 30m higher than the original height

3. V = 16.5m/s

4. V = 33m/s, twice the original velocity
h=10m
Unit VII WK4 Answers
W=Ek

5. Vbullet = 967m/s

6. a. Eint = 1106J
V=0m/s
V=?
Δx=0.85m
b. See notes
1200J
Child
Slide
Earth
Eg=mgh
94J
Ek=1/2mV2
WK4 Problem #6

A 24kg child descends a 5.0m high slide and reaches the
ground with a speed of 2.8m/s.


How much energy was dissipated due to friction in the process?
Do a pie chart analysis of the final state, using accurate % of the
pie to represent the amount Eint in the process.
Power


One of the events in the
“World’s Strongest Man”
competition is called Atlas
Stones. Five stones are
placed at the base of five
platforms.
Strength or power is judged
by the time it takes to
complete the task.
Work Energy
P

time
time
Units:
J/s = Watts
…also measured in units of horsepower (746W = 1hp)
PHHS Strong Student Competition

Find the total work done / energy exerted and the
amount of time to do it.
W Eg mgh
P 

t
t
t