Common Exam 2
During class (1 - 2:55 pm) on 6/14, Tue
Room: 412 FMH (classroom)
Bring scientific calculators
E
Exam
covers lectures
l t
after
ft common exam 11.
Review session: Monday during class
1
Last class
In 1D:
Kinetic Energy
Work
Net Work
Work-Energy Theorem
Today
Kinetic Energy, Work, Net Work, W-E Theorem in 2D
Conservative vs
vs. Non-conservative
Non conservative forces
Gravitational Potential Energy
2
1
For 2D motion…
Kinetic energy
K .E . =
1 2 1
mv = m ( vx2 + v y2 )
2
2
3
What if we have force and displacement with an angle?
Work in 2D
2
Work in 2D
What if force and displacement are perpendicular?
Example: Uniform circular motion
Fnet
Fnet
Fnet
No change in “magnitude” of velocity
No kinetic energy change
No work, sorry !
(Velocity does change, because the “direction” changes.)
Displacement
positive work
Force
Displacement
zero work
Force
Displacement
negative work
Force
Work vs.
Disp.-Force angle
Cos!
3
Work done by a constant force in 2D
W = F d cos θ
Define
F ,d
Scalar (dot) product
magnitude
Force
θ
Note:
0 ≤ θ < 90
θ = 90
90 < θ ≤ 180
≡ F ⋅d
Displacement
positive work
zero work
negative work
7
Example
Angle = 20 degree, F = 5 N, d = 2 m.
Find the work done by force F.
4
Math Review: Scalar (dot) product using components
d = (d x , d y )
F = ( Fx , Fy )
d cosθ
W = F
= Fxd
x
+ Fyd
Commutative property
Distributive property
F ,d
≡ F ⋅d
y
A⋅ B = B⋅ A
(A + B )⋅ C = A ⋅ C + B ⋅ C
9
iClicker Quiz
Force
F = (3N , − 2 N )
is acting on an object,
which makes a displacement
d = (2m, 4m)
Find the work done by this force.
A)
(6 J , - 8 J )
B)
32 + 22 × 22 + 42
C) 6 + 12 -4 -8 = 6 J
D) -2
2J
E) Not enough information. I need to know theta.
10
5
If multiple forces are applied on an object,
F2
F1
vi
Displacement
vf
d
F3
Wnet = Fnet id = ( F1 + F2 + F3 + ...)id
= F1 id + F2 id + F3 id + ... = W1 + W2 + W3 + ...
= K f − Ki =
1 2 1 2
mv f − mvi
2
2
Work-Energy Theorem in 2D
If multiple forces are applied on an object,
F2
vi
F1
Displacement
vf
d
F3
Wnet = W1 + W2 + ... = K f − K i
6
Example & iClicker
rope
F=T = 10 N
2 kg
θ = 30O
d=5m
Frictionless surface
1 Find the forces on the box.
1.
box
2. Find the work done by each force.
A) Positive
P iti
B) Zero
C) Negative
3. Find the net work
4. If initial velocity is 3 m/s, what is final kinetic energy?
Next topic
Conservative vs. Non-conservative forces
gy
Gravitational Potential Energy
14
7
Motivation
Why do we learn about potential energy and
conservation of mechanical energy?
15
iClicker Quiz
Work done by a gravity force
h1 = 4m
h3 = 3m
h2 = 2m
Mass =1 kg
+
-
h1 = 4m
Mass =1 kg
vs.
+
+
Fg = mg
h4 = 1m
(Work along h1->h
>h2->h
>h3->h
>h4)
Which work is greater?
vs
h4 = 1m
(Work along h1->h
>h4)
A) Work along h1->h2->h3->h4
B) They are equal
C) Work along h1->h4
D) Not enough information
16
8
iClicker Quiz
h1 = 3m
Work done by a gravity force
x1=0 m
x2=4 m
Mass =1 kg
vs.
Fg = mg
vs.
h4 = 0m
((Work along
g red arrows)) vs ((Work along
gg
green arrows)) vs ((work along
g blue arrow))
Which work is the greatest? A) Work along green arrows
B) Work along blue arrow
C) Work along red arrows
D) Work along red and green are equal
and greater than work along blue arrow
17
E) They are all equal
Conservative force
If the work done by a force depends only on initial and final positions,
the force is called a conservative force.
Gravity
y force is “a” conservative force.
18
9
iClicker Q:
Position
Work done by friction force
3m
2m
1m
-
-
vs.
x1 = 1m
-
4m
f k = 1N
x4 = 4m
: opposite to
displacement
-
In which case, the work by friction is more negative?
A)
B)
C)
D)
Top
Bottom
They are equal
N.E.I.
Is friction force a conservative force?
19
Non-Conservative force
If the work done by a force depends not only on initial and final
positions, but also on the path between them,
the force is called a non-conservative force.
Example: Friction force,Tension, normal force, and force applied by a
person.
20
10
Conservative forces in Phys 102:
Gravity, Spring force(to be studied in future)
Non-Conservative forces in Phys 102:
Friction, Normal force, Tension, Other applied forces
21
Mass=1 kg
height=10 m
Can you find the work done by
the gravity force along the
path?
A)) Yes,, of course.
B) Please… No.
C) Not enough information ☺
height=2 m
22
11
Gravitational Potential Energy
(Work done by gravity on object with mass m )
hi
= Wg = mg (hi − h f )
= mgh
g i − mgh
g f
Fg = mg
= U g ,i − U g , f
hf
Ug
where
U g = mgh
= Gravitational “Potential Energy", which depends on position only.
Energy
gy related to position,
p
which potentially
p
y converted to kinetic energy.
gy
Wg = U g ,i − U g , f = −(U g , f − U g ,i ) = −ΔU g
Work done by gravity is negative of gravity Potential Energy change
23
Gravitational Potential Energy
Mass m
Gravitational Potential Energy:
Height:
h
U g = mgh
(Note: Height, h, should be measured along vertical direction.)
The higher and the heavier an object is,
the g
greater the gravitational
g
potential
p
energy
gy is.
24
12
Example:Bungee Jump and Gravitational potential energy
(a) Is that, by any chance, …uh…
Prof. Ahn?
(b) Suppose that the person’s mass
is 70 kg and the height of the bridge
from the water surface is 120 m.
What is the person’s gravitational
potential energy when the person
starts to jump?
The height and the potential energy
is measured from the water surface.
New Zealand, 2004
initial
m= 3 kg
10 m
final
30 deg
iClicker Quiz
The change in gravitational potential energy is _________
(a) 3 x 9.8 x 10 J
(b) 3 x 9.8 x (-10) J
(c) something else
26
13
initial
m= 3 kg
10 m
final
30 deg
Example:
The change in gravitational potential energy is _______ J
27
initial
m= 3 kg
10 m
final
30 deg
Example:
The work done by gravity is _________
28
14
Wg = U g ,i − U g , f = −(U g , f − U g ,i ) = −ΔU g
Work done by gravity force is
negative of gravitational potential energy change
If gravitational force is the only force that does work,
, then , from Work-Energy theorem, Wnet
Wnet = Wg
= ΔK ,and above.
ΔK = −ΔU g
ΔK + ΔU g = 0
K + U g does not change.
29
Conservation of Mechanical Energy with gravity force
Define Mechanical energy :
1
2
2
Emech ≡ K + U g = mv + mgh
If
Wnet = Wg
or, if gravity force is the only forces that does work,
Emech , f = Emech ,i
: Mechanical energy does not change.
Mechanical energy
gy is conserved.
In general, if conservative forces are the only forces that do work,
mechanical energy is conserved.: “Conservation of mechanical energy”
30
15
Conservation of Mechanical Energy
If conservative forces only cause energy changes, the kinetic and potential energy
can change, but their sum, the mechanical energy Emec of the system, cannot change.
Example: Falling ball
h1
Less kinetic energy, but
M
More
potential
t ti l energy
v1
Height
h2
M
More
ki
kinetic
ti energy, b
butt
Less potential energy
v2
0
Conservation of Mechanical Energy
If conservative forces only cause energy changes, the kinetic and potential energy
can change, but their sum, the mechanical energy Emec of the system, cannot change.
Example: Falling ball
h1
Mechanical energy
v1
Height
Emech =
1 2
mv + mgh
2
Conservation of mechanical energy
h2
v2
1 2
1
mv1 + mgh1 = mv22 + mgh2
2
2
0
16
iClicker Quiz
A person throws a ball 30 degree from horizontal
from the top of a 20 m high building. Neglect the air
resistance.
True or false?
If he throws the ball at a different angle but with
the same speed, the ball would hit the ground
at a different speed.
(a) True
(b) False
33
Example: Pendulum
2 m rope
30 degree
V_i=0
What is the speed at the bottom?
34
17
Example. A car with its engine off costs along a straight highway,
which goes uphill. How far along the highway will the car go before
its stops, if its initial speed was 80 km/h, and the slope is 15o? The
tires roll, not skid.
35
18