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Midterm this Friday November 8
Classical Mechanics
Midterm 2 Review
Force and Energy







Newton’s Laws of Motion
Forces and Free Body Diagrams
Friction
Work and Kinetic Energy
Conservative Forces and Potential Energy
Work and Potential Energy
Knowledge of Units 1-3 will be useful
Mechanics Lecture 9, Slide 1
Midterm Exam


Multiple choice…but show your work and justification.
Calculations





Forces and Free-Body Diagrams

Weight and Pulleys

Springs

Friction

Gravitational

Normal
Work Calculations
Conservation of Energy
Conceptual questions…like checkpoint problems.
Bring calculators and up to five sheets of notes
Mechanics Lecture 8, Slide 2
Example Problem : Block and spring
A 2.5 kg box is held released from rest 1.5 m above the ground and slides
down a frictionless ramp. It slides across a floor that is frictionless, except for
a small section 0.5 m wide that has a coefficient of kinetic friction of 0.2. At
the left end, is a spring with spring constant 250 N/m. The box compresses
the spring, and is accelerated back to the right.
What is the speed of the box at the bottom of the ramp?
What is the maximum distance the spring is compressed by the box?
Draw a free-body diagram for the box while at the top of the incline ? When the spring
is maximally compressed? When the box is sliding on the rough spot to the right?
What is the total work done by friction? Each way?
What height does the box reach up the ramp after hitting the spring once?
Where will the box come to rest?
2.5 kg
k=250 N/m
h=1.5 m
mk = 0.4
d = 0.50 m
Mechanics Review 2 , Slide 3
Determining Motion
Force

Unbalanced Forces  acceleration
(otherwise objects velocity is constant)
Energy

Total Energy  Motion, Location


F  a
 F
a
m


F12  F21

Emechanical  K  U
Emechanical  Wnonconservative

rf

Determine Net Force acting on
object
Work

Wnet


  Fnet  dl  K

Conservative forces r0
U  Wnet  K
 Emechanical  U  K  0

Motion from Energy conservation
Emechanical, final  K f  U f

Use kinematic equations to
determine resulting motion
v f  at  v0 ;...
Emechanical, final  K i  U i  Emechanical
K f  K i  U i  Emechanical  U f
2
K i  U i  Emechanical  U f 
vf 
m
Mechanics Lecture 8, Slide 4
Unit 4: Main Points
Mechanics Lecture 4, Slide 5
Unit 4: Main Points
Mechanics Lecture 4, Slide 6
Unit 4: Main Points
Mechanics Lecture 4, Slide 7
Unit 5: Main Points
Mechanics Lecture 5, Slide 8
Unit 5: Main Points
Mechanics Lecture 5, Slide 9
Unit 5: Main Points
Mechanics Lecture 5, Slide 10
Inventory of Forces
 Weight
 Normal Force
 Tension
 Gravitational
 Springs
 …Friction
Mechanics Lecture 5, Slide 11
Mechanics Lecture 5, Slide 12
1) FBD
m2
N
f
m2
T
g
T
m2g
m1
m1
m1g
Mechanics Lecture 6, Slide 13
1) FBD
2) SF=ma
m2
N
T
m2
f
g
T
m2g
N = m2g
T – m m2g = m2a
m1g – T = m1a
m1
m1
m1g
add
m1g – m m2g = m1a + m2a
a=
m1g – m m2g
m 1 + m2
Mechanics Lecture 6, Slide 14
1) FBD
2) SF=ma
m2
N
f
m2
T
g
T
m1
m1
m2g
m1g
a=
m1g – m m2g
m1 + m2
m1g – T = m1a
T = m1g – m1a
T is smaller when a is bigger
Mechanics Lecture 6, Slide 15
Force Summary
Mechanics Lecture 5, Slide 16
Unit 6: Main Points
Mechanics Lecture 6, Slide 17
Unit 6: Main Points
Mechanics Lecture 6, Slide 18
Friction
Mechanics Lecture 6, Slide 19
Friction
Mechanics Lecture 6, Slide 20
Block
1 2
2x
at  a  2
2
t
F
mg sin   f k
a  net 
m
m
2x 

f k  mg sin   ma  m g sin   2 
t 

x 
Mechanics Lecture 5, Slide 21
Pushing Blocks

F23net
a
Fh1
F
 h1
(m1  m2  m3  m4 ) 4m1
F23netx  (m3  m4 )a  2m1a 
2m1 Fh1 Fh1

4m1
2
F23net y  N  (m3  m4 ) g  2m1 g
2
F23net  F
2
23net x
F
2
23net y
F 
2
  h1   2m1 g 
 2 
Mechanics Lecture 5, Slide 22
Work-Kinetic Energy Theorem
The work done by force F as it acts on an object that
moves between positions r1 and r2 is equal to the
change in the object’s kinetic energy:
But again…!!!

r2
W  K
 
W   F  dl

r1
1 2
K  mv
2
Mechanics Lecture 7, Slide 23
Unit 7: Main Points
Mechanics Lecture 7, Slide 24
Main Points
Mechanics Lecture 7, Slide 25
Summary Unit 8: Potential & Mechanical Energy
Mechanics Lecture 8, Slide 26
The Dot Product
Mechanics Lecture 7, Slide 27
Vectors!!!
Mechanics Lecture 8, Slide 28
Main Points
Mechanics Lecture 7, Slide 29
Mechanics Lecture 8, Slide 30
Unit 9:Main Points
Mechanics Lecture 8, Slide 31
Main Points
Mechanics Lecture 8, Slide 32
Energy Conservation Problems in general
For systems with only conservative forces acting
Emechanical  0
Emechanical is a constant
Emechanical  K i  U i  K f  U f  K (t )  U (t )
Mechanics Lecture 8, Slide 33
Example Problem
 
Wtension   T  dl  Tx
Wnet  Wtension  W friction  K
W friction  Wtension  K

1
K  m v 2f  v02
2


1
W friction  Wtension  m v 2f  v02
2

Mechanics Lecture 8, Slide 34
Example Problems
Emechanical  W friction
W friction   m k mgx
Emechanical, final  Emechanical,initial  W friction
mgh  mghi  m k mgx  mg hi  m k x 
h  hi  m k x 
Mechanics Lecture 8, Slide 35
Example Problems
Mechanics Lecture 8, Slide 36
Example Problems
Mechanics Lecture 8, Slide 37
Example: Pulley and Two Masses
A block of mass m1 = 1 kg sits atop an inclined plane of angle
θ = 20o with coefficient of kinetic friction 0.2 and is connected
to mass m2 = 3 kg through a string that goes over a massless
frictionless pulley. The system starts at rest and mass m2 falls
through a height H = 2 m.
Use energy methods to find the velocity of mass m2 just
before it hits the ground?
θ
m2
HH
=2
m
=2
2m
kg
Mechanics Lecture 19, Slide 38
Example: Block and spring
A 2.5 kg box is held released from rest 1.5 m above the ground
and slides down a frictionless ramp. It slides across a floor that
is frictionless, except for a small section 0.5 m wide that has a
coefficient of kinetic friction of 0.2. At the left end, is a spring
with spring constant 250 N/m. The box compresses the spring,
and is accelerated back to the right.
What is the speed of the box at the bottom of the ramp?
What is the maximum distance the spring is compressed by the
box?
2.5 kg
k=250 N/m
h=1.5 m
mk = 0.4
d = 0.50 m
Mechanics Review 2 , Slide 39
Example: Pendulum
v v 22ghgh
hh
Conserve Energy from initial to final position.
1 2
mgh  mv
2
v  2 gh
Mechanics Review 2 , Slide 40
Example Problems
Mechanics Lecture 8, Slide 41
Example Problems
Mechanics Lecture 8, Slide 42
Gravitational Potential Energy
 
r  rE

rE

r
 
r  rM

rM
Mechanics Lecture 8, Slide 43
Gravitational Potential Problems
 
r  rE

rE
 
r  rM
 conservation of mechanical energy
can be used to “easily” solve
problems.
Emechanical  K  U 

r

rM
 Add potential energy from each
source.
GM E m
U Earth (rE )  
rE
U Moon (rM )  
1
mv(h) 2  U (h) gravity
2
 Define coordinates: where is
U=0?
U (r )  
GM E m
 0 as r  
r
GM M m
rM

GM E m GM M m
U total (r )       
r  rE
r  rM
Mechanics Lecture 8, Slide 44
Trip to the moon
Ki  U i  K f  U f
1 2 GM E m GM m m 1 2 GM E m GM m m
mvi 

 mv f 

2
RE
d M E
2
d E M
RM
 1 2 GM E GM m GM E GM m 

v f  2 vi 



RE
d M E d E M
RM 
2
Mechanics Lecture 8, Slide 45