Announcements:
1 Take your 1st quiz back
1.
2. 2nd Quiz will be on Friday
3. This
h week
k H
HW will
ll b
be due
d
on coming Wed. instead of
Mon.
Mon
Power
• The rate at which work is done by a force is power.
power
• Average power is work W done in time Δt
W
Pave =
Δt
• The instantaneous rate of doing work (instantaneous power)
dW
P=
dt
• Units: Watt [W]
1 W = 1J/s
1 horsepower = 1 hp = 746 W
1 kW-hour
kW h
= 3.6
3 6 MJ
• Power from the time-independent
time independent force and velocity:
(
)
d ( x)
dW d F ⋅ x
=
=F⋅
= F ⋅v
P=
d
dt
dt
d
dt
d
Instantaneous power !
Crate of Cheese
VERY S
V
SIMILAR
TO HW # 6
An initially stationary crate of cheese (mass m) is pulled via a cable a
distance d up a frictionless ramp of angle θ where it stops.
(a) How much work WN is done on the crate by the Normal during the
lift?
(b) How much work Wg is done on the crate by the gravitational force
during the lift?
(c) How much work WT is done on the crate by the Tension during the
lift?
((d)) If the speed
p
of the moving
g crate were increased,, how would the above
answers change? What about the power?
•Chapter 7: What happens to the KE of when work is done on it.
[ KE: “energy of motion”
W: energy transfer via force ]
•Chapter 8: Potential energy ??
Conservative vs non
non-conservative forces
• Can you get back what you put in?
• What happens when you reverse time?
Win = -Wout
•Net work done by a conservative force on a object moving around every
closed path is zero.
b
Wab = ∫ F (x) • dx
a
Wab,1 = Wab,2
Conservative forces
- gravitational
it ti l force
f
- spring force
&
Wab,1 = −Wba,2
Non-conservative forces
- kinetic frictional force
(noise, heat,…)
Potential energy: Energy U which describes the configuration (or spatial arrangement)
of a system of objects that exert conservative forces on each other. It’s the stored
energy in system.
ΔU = −W
You can always
the
zero (reference value)
Gravitational Potential energy: [~associated with the state of separation]
yf
ΔU grav = − ∫ (−mg)dy = mg(y f − yi ) = mgΔy
yi
If U grav (y
( = 0) ≡ 0
th
then
U grav (y)
( ) = mgy
ΔUggrav ↑ if going up
ΔUgrav ↓ if going down
Elastic Potential energy: [~associated with the state of compression/tension
of elastic object]
2
2
1
1
ΔU spring = 2 kx f − 2 kx i
If
U spring (x
( = 0) ≡ 0
then
U spring (x)
( ) = 12 kx
k 2
ΔUspring ↑ if x goes
↑ or ↓ (any displacement)
Don’t forget…
Work done by force (general):
Work by Gravitational force:
Work by Spring force:
Work-Kinetic Energy theorem:
W = ∫ F ( x ) • dx = F • d
Wg = Fg • d
Wspring = 12 kx
k i2 − 12 kx
k 2f
Wnet = ΔKE = KE f − KEi
Potential energy (if conservative force):
ΔU grav = mgΔy
2
2
1
1
ΔU spring
=
kx
−
kx
i
f
i
2
2
W = −ΔU
ΔU
If U grav (y = 0) ≡ 0
then U grav (y) = mgy
If U spring (x = 0) ≡ 0
then U spring (x) = 12 kx 2
Mechanical energy:
•
E mech = KE + U
Only conservative forces (gravity & spring) cause energy transfer (work)
Wnet = ΔKE
W = −ΔU
•
Sec. 7- 3
Sec. 8-1
Isolated system: Assuming only internal forces (no external forces yet - Sec. 8.6)
– No
N external
t
l force
f
from
f
outside
t id causes energy change
h
inside
i id
ΔE mech = 0 = Δ(KE + U )
ΔKE = −ΔU
ΔU
How to solved these problems
My preference is to always use Mechanical Energy
E mec ( ffinal)) = Emec ((initial))
K f + U f = K i + Ui
When Friction is present, it always removes energy,
so E(final) is less the E(init
E(initioa
ioal)
E mec ( final ) = Emec (initial ) − Energy removed by work of friction
K f + U f = K i + U i − Ff • d
Always
y think about the pproblem and the signs!!
g
Gravitational Potential Energy
Conservative Force Problems
Example of Conservation of Mechanical Energy
Consider a 100 kg bobsled starting with vo = 0 on a frictionless track. Compute the
KE, PE, and total E at the labeled points. Zero of potential at the bottom.
1 2
KE = mv
2
PE = mgh
Emech=KE+PE
KE
PE
Emech = KE + PE
Height
g h
600 m
588,000 J
0J
588,000 J
588,000
,
J
196,000
,
J
392,000
,
J
400 m
588 000 J
588,000
392 000 J
392,000
196 000 J
196,000
200 m
588 000 J
588,000
588 000 J
588,000
0J
0m
Example of Conservation of Mechanical Energy
Consider a 100 kg bobsled starting with vo = 0 on a frictionless track. Compute the
KE, PE, and total E at the labeled points. Zero at the top!
1 2
KE = mv
2
PE = −mgh
Emech=KE+PE
KE
PE
Emech = KE + PE
Height
g h
0m
0J
0J
O
0 JJ
-200 m
0J
196 000 J
196,000
-196 000 J
-196,000
00 JJ
392
392,000
392000
392,000
000J
400 m
-392,000
392 000 J -400
0J
588,000 J
-588,000 J -600 m
Question
Three identical projectiles are fired at different launch angles
and with different initial velocities. Which projectile has the
greatest potential energy when it’s at the peak in its
t j t ?
trajectory?
1. A
2. B
3. C
4. All are equal
Can you tell me anything about the kinetic energies?
Question
A block initiall
initially at rest slides down
do n a frictionless ramp and attains a
speed of v at the bottom. To achieve a speed of 2v, how many times
as high
g must a new ramp
p be?
KE f = KEi + PEi − PE f = 0 + mgh
1
2
mv2f = mgh
v f = 2gh
1.
2
2.
3.
4.
5.
6.
1
2
3
4
5
6
Question
A young girl wishes to select one of the frictionless playground
slides below to give her the greatest possible speed when she
reaches the bottom of the slide. Which one should she choose?
B
A
1.
2.
3.
4.
5.
A
B
C
D
Any of them
C
D