Download AP B Chapter 6 Work and Energy

Conservation of Energy and Momentum
Three criteria for Work
There must be a force.
 There must be a displacement, d.
 The force must have a component
parallel to the displacement.
Work, W = F x d, W = Fd cos θ.

F
F
x

If a Force does not affect displacement, it
does NO WORK.
FA
Fg

The Earth exerts a force on the case, but does
no work. The man exerts a horizontal force &
does work.
Ex 1:Work on a backpack

Determine the work a hiker must do on a 15.0kg
backpack to carry it up a hill of height h=10.0m.
d
θ
h
h = d cosθ
Soln: In the vertical direction, if up is positive, ΣFy
= may If mg is the force of gravity down, and Fh is
the hiker’s force up, then ΣFy = Fh - mg = 0
(assuming negligible acceleration) so Fh = mg =
147N.
 Work done by the hiker, Wh = Fh (d cos θ) = Fh h
= mgh = (147N )(10.0m) = 1470 J

Does Earth do Work on the Moon?
last (third)quarter
waning Moon
moon orbit`s
earth
v
SUN
gibbous moon
crescent
Fg
earth
full moon
new moon
gibbous moon
crescent
waxing Moon
first quarter
The angle between the Force and instantaneous displacement θ is 90. Cos 90=0,
therefore NO WORK is done by gravity. (W=Fdcosθ)
Kinetic Energy
Regardless of the types of Energy, in any process
it is CRUCIAL to understand the total energy
remains the same after the process as before!
 Energy traditionally is defined as ‘the ability to do
work or cause change’. (change = acceleration)
 Kinetic Energy is energy of motion. A moving
object has the ability to do work.
 KE = ½ mv2 and is measured in Joules, J.

Do Work!
Doing work on an object causes a change in the
object’s energy… More work = more change.
 Wnet = KE2 – KE1 so Wnet = ΔKE
 The net work done on an object is equal to the
change in its kinetic energy.
 This is the Work-Energy principle.

Work to stop a car

An automobile traveling 60 km/h can
brake to stop within a distance of 20m.
If the car is going twice as fast, what is
its stopping distance?
Solution
The stopping force, F is about constant. The
work needed to stop the car, Fd, is
proportional to the stopping distance. Using
the Work-Energy theorem we note that F and d
are in opposite directions:
 Wnet = Fd(cosθ) = Fdcos180= -Fd
 Wnet = ΔKE = 0 – ½ mv2
 Since force & mass are constant, stopping
distance increases as v2
 If v is doubled, stopping distance is 22 as great
or 20m(4) = 80m.

Your Turn to Practice
Please do Ch 6 Rev p 174 #s 1-6
 Pg 175 #s 17-20

Potential Energy
Energy stored in an object due to its
position or condition is Potential Energy.
 Common types of Potential energy include:

 Gravitational PE – energy due to position above
Earth.
 Elastic PE- energy stored due to work being
done on an object and the object restoring its
original shape.
Gravitational PE
Gravitational Potential Energy results from
lifting an object to a height, h above some
reference point (like the ground).
 PEgrav =mgy (where y is the height above y0 =
0m).
 Work done to lift a mass from y1 to y2 is equal
to the change in PE from y1 to y2.
 Just like KE, though we may not know the
original amount of energy an object has, if we
do WORK on it, we know the change in energy.
W = mg (y2 – y1 ) = ΔPE.

It is important to note, CHANGES in PE
depend only on vertical height, h, and
not on the path taken to get there…
Recall the hiker and backpack? How
about the Pyramids of Ancient Egypt?
 Does the horizontal part of stairs play a
role in your PE as you climb them?

Elastic Potential Energy
If you push (or pull) on a spring (you do work to
compress or stretch it), the spring can store
energy and can do work when it is released.
 To hold a spring, stretched or compressed, a
distance, X from its normal un-stretched length,
requires a force, Fp that is directly proportional
to X.
 Fp = kx where k is a spring constant based on
stiffness of the spring material.

Hooke’s Law
x
A stretched or compressed spring,
wants to restore itself in the opposite
direction. This restoring force is the
spring’s force. This is Hooke’s Law or
the Spring equation:
 Fs = - kx, where k is the spring constant
and the negative sign is because the
spring exerts the force in a direction
opposite the displacement.

F
m
Compressing or Stretching a Spring
Initially at Rest:
Two forces are always
present: the outside
force Fout ON spring
and the reaction force
Fs BY the spring.
x
x
m
m
Compressing
Stretching
Compression: Fout does positive work and Fs does negative
work (see figure).
Stretching:
Fout does positive work and Fs does negative
work (see figure).
General Case for Springs:
If the initial displacement is not zero, the work done is
given by:
Work
1
2
2
2
kx
1
2
2
1
kx
F
x1
x2
m
x1
x2
m
Stretch a spring



As you pull on a spring, the Force varies with the stretch… (to
stretch twice as far does NOT require twice the force!)
Potential energy of a spring is found considering Work done on
it to stretch it.
Consider AVERAGE FORCE to stretch a spring from 0 to x
from Fp = kx, we get
F
W

1
0
2
F px
kx
1
kx
2
1
kx
2
x
1
2
kx
2
Elastic Potential Energy = ½ kx2 (x = displacement
stretched or compressed from normal equilibrium
position.)
Conservation of Mechanical Energy
E is the total mechanical energy of a
system. E is the sum of kinetic and
potential energies at any moment.
 E = KE + PE
 KE2 + PE2 = KE1 + PE1 OR:
 E2 = E1 = constant
 Therefore: ΔPE = -ΔKE or if KE
increases, then PE must decrease.

Problem solving with Energy conservation
A rock at height h is dropped
from rest. (KE = 0). As it falls,
PE decreases, but it gains
speed (KE increases = to the
change of PE)
 At any point, the total
mechanical energy is given
by:
 E = KE + PE = ½ mv2 + mgy
where v is the velocity at
height y…

1
2
3
Total Mechanical energy at 2 different
points of fall must be equal.
1 2
mv1 mgy1
2
1 2
mv2 mgy2
2
Conservation of a spring

A dart of mass 0.100kg is pressed
against the spring of a toy dart gun. The
spring, k = 250N/m, is compressed
6.0cm and released. If the dart
detaches from the spring when the
spring reaches equilibrium, what speed
does the dart acquire?
1 2
mv1
2
1 2
kx1
2
1 2
mv2
2
1 2
kx2
2
In the Horizontal direction, the only force on the dart is the force exerted
by the spring. Vertically, gravity is balanced by the normal force on the
gun barrel. (After the dart leaves the gun, it follows the projectile path
under gravity) Use the above equation w position 1 being at the maximum
compression of the spring, so v1 = 0 (dart not released yet) and x1 = 0.06m. Point 2 we choose to be the instant the dart flies off the end of the
spring, so x2 = 0 and we want to find v2.
We can rewrite the equation:
0
v
2
2
v2
1 2
kx1
2
kx12
m
v22
1 2
mv2 0
2
(250N / m)(0.06m) 2
(0.100kg)
3.0m / s
m2
9.0 2
s
Law of Conservation of Energy

The total energy is neither increased nor
decreased in any process. Energy can
be transformed from one form to
another, and transferred from one body
to another, but the total amount remains
constant.
Power

Power is the rate at which work is done,
or the rate at which energy is
transformed.
work energytransformed J
P
Watt
time
time
s
Another unit of power is the
horsepower (hp). 1 hp = 746 W.
Your turn to Practice
Please do Ch 6 Rev p 175 #s 21, 22, 29,
30, 31
 P 177 #s 58, 59, 60.
