Physics I
Review 2
Review Notes
Exam 2
Rev. 26-Oct-04 GB
R2-1
Work
 Work is a measure of the energy that a force puts into (+) or
takes away from (–) an object as it moves.
 We will see that work is a useful way to solve problems where
the force on an object is a known function of position.
 Example: the force of an object connected to an ideal spring:


F  k x

x
where is the displacement from equilibrium. (Hooke’s Law)
R2-2
Work for Constant Force
 
W  Fd
 
W  F d cos( )
R2-3
Vector Dot Product
F

d
If you know lengths and angle:
 
W  F d cos( )
If in opposite directions:
W  Fx d x  Fy d y  Fz d z 
 
W Fd
 
W  F d
If at right angles:
W0
If you know components:
If in the same direction:
R2-4
Work for Variable Force
W   F ( x )dx
xf
xi
(This is the version for one dimension.)
R2-5
Work-Energy Theorem
 Net work is done on an object by the net force:
xf
Wnet   Fnet dx
xi
 Kinetic energy defined for an object:

2
2
2
2
K  12 m v  12 m v x  v y  v z

 Work-Energy Theorem: (without proof)
K f  K i  Wnet
R2-6
Class #10
Take-Away Concepts
1.
Work is a measure
  ofenergy added to (+) or taken away (–).
W  F  d  F d cos( ) (constant force)
xf
W   F dx
(variable force, 1D)
xi
2.
Vector dot product defined.

2 1
2
2
2
3. Kinetic Energy: K  m v  2 m v x  v y  v z
4. Work-Energy Theorem: K f  K i  Wnet
1
2
5.
6.
7.

Positive net work means an object’s K.E. increases (speeds up).
Negative net work means an object’s K.E. decreases (slows down).
Zero work means an object’s K.E. stays constant (constant speed).
R2-7
Does work depend on the path?
Conservative Forces
For general forces, the work does depend on the path that we take.
However, there are some forces for which work does not depend on
the path taken between the beginning and ending points.
These are called conservative forces.
A mathematically equivalent way to put this is that the work done by
a conservative force along any closed path is exactly zero.


F

d
x
 cons  0
(The funny integral symbol means a path that closes back on itself.)
R2-8
Conservative Forces
Non-Conservative Forces
Examples of Conservative Forces:
 Gravity
 Ideal Spring (Hooke’s Law)
 Electrostatic Force (later in Physics 1)
Examples of Non-Conservative Forces:
 Human Pushes and Pulls
 Friction
R2-9
Conservative Forces
and Potential Energy
If we are dealing with a conservative force, we can simplify the process
of calculating work by introducing potential energy.
1. Define a point where the potential energy is zero (our choice).
2. Find the work done from that point to any other point in space.
(This is not too hard for most conservative forces.)
3. Define the potential energy at each point as negative the work done
from the reference point to there. Call this function U.
4 The work done by the conservative force from any point A to any
point B is then simply W = U(A)–U(B).
R2-10
Two Common Potential Energy
Functions in Physics 1
Gravitational Potential Energy
U g  m g (y  y0 )  m g h
(y0 is our choice to make the problem easier)
Spring Potential Energy
U s  12 k ( x  x 0 ) 2
(x0 is the equilibrium position and k is the spring constant)
R2-11
Potential Energy, Kinetic Energy,
and Conservation of Energy
Recall the Work-Energy Theorem:
 K  Wnet
And for conservative forces we have
Wcons   U
If the non-conservative forces are zero or negligible, then
Wnet  Wcons
Putting it together,
 K   U
or
K U  0
Another way to say this is the total energy, K+U, is conserved.
R2-12
Example Problem
Skateboarder Going Up a Ramp
K0
U  mgh
K  12 m v 2
U0
d
v
h

m v2  0  0  m g h
v2
h
v2
h
d

2g
sin( ) 2 g sin( )
1
2
R2-13
Class #11
Take-Away Concepts
1.
Multi-dimensional
form of work integral:

xf
 
W   F  dx

xi
2.
3.
Conservative force = work doesn’t depend on path.
Potential Energy defined for a conservative force:
 
U ( A )    F  dx
A
0
Ug  m g ( y  y0 )  m g h
U s  12 k ( x  x 0 ) 2
4.
Gravity:
5.
6.
Spring:
Conservation of energy if only conservative forces operate:
 K    U or  K   U  0
R2-14
Is Mechanical Energy
Always Conserved?
Total Mechanical Energy
EKU
E  K  U  0 if only conservative forces act
When Non-Conservative Forces Act
E  K  U  Wnon cons
This is equivalent to
K f  U f  K i  U i  Wnon cons
Non-conservative forces add (+) or subtract (–) energy.
R2-15
Example of Energy Lost to
Friction (Non-Conservative Force)
Kf  0
U f  m g h  70  9.8  1.5  1029J
K i  12 m v 2  12  70  64  2240J
Ui  0
v

d
h
h  d sin( )  3  12  1.5m
A skateboarder with mass = 70 kg starts up a
30º incline going 8 m/s. He goes 3 m along the
incline and comes to a temporary stop. What
was the average force of friction (magnitude)?
R2-16
Example of Energy Lost to
Friction (Non-Conservative Force)
Kf  0
U f  m g h  70  9.8  1.5  1029J
K i  12 m v 2  12  70  64  2240J
Ui  0
v
d
h

Wfriction  K f  U f  K i  U i  1029  2240  1211J
Wfriction  1211
Ffriction ,avg 

 403.7 N
d
3
What do the – signs mean?
R2-17
Elastic and Inelastic Collisions
Momentum is conserved when the external forces are zero or so
small they can be neglected during the collision. This is often true.
In many collisions a large percentage of the kinetic energy is lost.
These are known as inelastic collisions. For example, any collision
in which two objects stick together is always inelastic.
If the kinetic energy after a collision is the same as before, then we
have an elastic collision. During the collision, some of the kinetic
energy can convert to potential energy of various kinds, but after the
collision is over all of the kinetic energy is restored.
R2-18
Elastic Collisions in
One Dimension
+X
Initial
v1i
Final
v1f
v2i
v2f
Conservation of Momentum:
m1v1i  m 2 v 2i  m1v1f  m 2 v 2 f
Conservation of Energy:
1
2
m1v 21i  12 m 2 v 2 2i  12 m1v 21f  12 m 2 v 2 2 f
Two equations, two unknowns (final velocities).
R2-19
Elastic Collisions in
One Dimension
+X
Initial
Final
v1i
v2i
v1f
v2f
m1  m 2
2 m2
v1f 
v1i 
v 2i
m1  m 2
m1  m 2
2 m1
m 2  m1
v 2f 
v1i 
v 2i
m1  m 2
m1  m 2
R2-20
Elastic Collisions in
One Dimension - Example
m1  2 m 2
v1i  0
v 2i  1 m/s
+X
v2i
v1f
2 1
2
2
v1f 
( 0) 
(1)   m/s
2 1
2 1
3
2
1 2
1
v 2f 
( 0) 
(1)   m/s
2 1
2 1
3
v2f
Initial: m2 has all (–1) of the
mom. and KE.
Final: m1 has –4/3 of the mom.
and 8/9 of the KE.
m2 has +1/3 of the mom.
and 1/9 of the KE.
R2-21
Class #12
Take-Away Concepts
1.
Modification of energy conservation including nonconservative forces:
 E   K   U  Wnon cons
2.
3.
4.
Non-conservative work adds (+) or subtracts (–)
energy from the system.
Elastic collision preserves KE before and after.
(Don’t assume all collisions are elastic, most are not.)
Special equations for 1D elastic collisions.
m1  m 2
2 m2
v1i 
v 2i
m1  m 2
m1  m 2
2 m1
m  m1
v 2f 
v1i  2
v 2i
m1  m 2
m1  m 2
v1f 
R2-22
Definitions
Angular Position:
Angular Displacement:

 (in radians)
 
    0
Average or mean angular velocity
isdefined as follows:

  0 

avg 

t  t 0 t
Instantaneous angular velocity
 or just
 “ angular velocity”:
 d 

  lim

t  0 t
dt
Wait a minute! How can an angle have a vector direction?
R2-23
Direction of Angular Displacement
and Angular Velocity
•Use your right hand.
•Curl your fingers in the direction
of the rotation.
•Out-stretched thumb points in
the direction of the angular
velocity.
R2-24
Angular Acceleration
Average angular acceleration is defined as follows:

 avg
 

  0 


t  t0
t
Instantaneous angular acceleration or just “angular
acceleration”:



 d  d 2 

  lim

 2
t  0 t
dt dt
The easiest way to get the direction of the angular acceleration is
to determine the direction of the angular velocity and then…
 If the object is speeding up, angular velocity and acceleration
are in the same direction.
 If the object is slowing down, angular velocity and acceleration
are in opposite directions.
R2-25
Equations for Constant 
1.
  0  t  t 0 
2.    0  0 ( t  t 0 )  2  ( t  t 0 )
1
1




(0  )( t  t 0 )
3.
0
2
4.    0  ( t  t 0 )  2  ( t  t 0 )
1
5.
2
x
v
a
2
2  02  2  0 
R2-26
Relationships Among
Linear and Angular Variables
MUST express angles in radians.
s  r
v  r
a tangential   r
a centripetal
v 2 2 r 2


 2 r
r
r
The radial direction is defined to be +
outward from the center.
a radial  a centripetal
R2-27
Energy in Rotation
Consider the kinetic energy in a rotating object. The center of
mass of the object is not moving, but each particle (atom) in the
object is moving at the same angular velocity ().
K   12 m i v i   12 m i 2 ri  12 2  m i ri
2
2
2
The summation in the final expression occurs often when
analyzing rotational motion. It is called the moment of inertia.
R2-28
Moment of Inertia
For a system of discrete “point” objects:
I   m i ri
2
For a solid object, use an integral where  is the density:
I    r 2 dx dy dz
We may ask you to calculate the moment of inertia for point objects, but we will
give you a formula for a solid object or just give you its moment of inertia.
I for a solid sphere:
I for a spherical shell:
I  52 M R 2
I  23 M R 2
R2-29
Correspondence Between
Linear and Rotational Motion
x
v
a
mI
F
K  12 I 2


  I
You will solve many rotation problems
using exactly the same techniques you
learned for linear motion problems.
R2-30
Class #13
Take-Away Concepts
1.
2.
Definitions of rotational quantities: , , .
Centripetal and tangential acceleration.
 mi ri1 2
4. Rotational kinetic energy: K  2 I 


5. Introduction to torque:   I 
3.
Moment of inertia: I 
6.
Correspondence
x
mI
v
F
2
a
R2-31
Review of Torque
For linear motion, we have “F = m a”. For rotation, we have


  I
The symbol “” is torque. We will define it more precisely today.


 When the rotation is speeding up, and are in the
same direction.


 When the rotation is slowing down, and are in
opposite directions.
Torque and angular acceleration are
always in the same direction in Physics 1.
R2-32
The Vector Cross Product
We learned how to “multiply” two vectors to get a scalar.
That was the “dot”
 product:



d  a  b | a | | b | cos( )
Now we will “multiply”
two vectors
to get another vector:




 
c  a  b ; | c |  | a | | b | sin( )
The direction comes from the right-hand rule. It is at a right angle

to the plane formed by a and b. In other words, the cross product

is at right angles to both a and b . (3D thinking required!)
R2-33
The Vector Cross Product
R2-34
Torque as a Cross Product
  
  rF
 

|  |  | r | | F | sin( )

r is the vector from the axis of rotation to where the force is applied.
The torque can be zero in
 three different ways:
1. No force is applied (| F |  0 ).

|
2. The
 force is applied at the axis of rotation ( r |  0 ).

3. F and r in the same or opposite directions (sin( )  0 ).
R2-35
Angular Momentum of a Particle
center of rotation (defined)

r


p  mv
Angular momentum of a particle
once a center is defined:
  
l  r p
(What is the direction of angular
momentum here?)
Once we define a center (or axis) of rotation, any object with a
linear momentum that does not move directly through that point
has an angular momentum defined relative to the chosen center.
R2-36
Class #14
Take-Away Concepts




I

1. : Speeding up, slowing down.
.
2.
Definition of vector cross product:
 
   
c  a  b ; | c |  | a | | b | sin( )
  
3. Torque as a cross product:   r F .
 
4. Angular momentum of a particle: l  r  p .
R2-37
How Does Angular Momentum of
a Particle Change with Time?
Take the time
 derivative of angular momentum:


d l d   d r   dp
 ( r  p) 
p  r 
dt dt
dt
dt
Find each term separately:
so

dr   
 p  v  p  0 (Why?)
dt

 dp  

r
 r  Fnet  net (Why?)
dt

dl 
 net (Newton’s 2nd Law for angular momentum.)
dt
R2-38
Angular Momentum of a Particle:
Does It Change if  = 0?
Y
(0,0)

r

r
X
(red)
(blue)
(0,–3)
(4,–3)


p  m v = 1 kg m/s (+X dir.)
The figure at the left shows the same
particle at two different times. No forces
(or torques) act on the particle.
Is its angular momentum constant?
(Check magnitudes at the two times.)
Blue angle:  = 90º
l = r p sin() = (3) (1) sin(90º) = 3 kg m2/s
Red angle:  = arctan(3/4) = 36.87º
l = r p sin() = (5) (1) sin(36.87º) = 3 kg m2/s


[r sin()] is the component of r at a right angle to p . It is
constant.
It is also the distance at closest approach to the center.
R2-39
Conservation of
Angular Momentum
Take (for example)
two rotating objects that interact.

d l1 

 on 1 from 2  ext on 1
dt
d l2 

 on 2 from 1  ext on 2
dt
The total angular
momentum


 is the sum of 1 and 2:
d L d l1 d l2 



 ext on 1  ext on 2 (Why?)
dt
dt
dt
If there are no external torques, then
dL
0
dt
R2-40
Class #15
Take-Away Concepts
  
1. Angular momentum of a particle (review): l  r  p .
2.
Newton’s 2nd Law for angular momentum:
3.
Conservation of angular momentum (no ext. torque):

dl 
 net
dt

dL
0
dt
R2-41
Formula Sheet Organization
Linear Kinematics
1.
v  v 0  a t  t 0 
2.
x  x 0  v 0 ( t  t 0 )  12 a ( t  t 0 ) 2
3.
x  x 0  12 ( v 0  v )( t  t 0 )
4.
x  x 0  v( t  t 0 )  12 a ( t  t 0 ) 2
5.
v 2  v 02  2a x  x 0 
R2-42
Formula Sheet Organization
Newton’s 2nd Law and Linear Momentum
 

F

F

m
a
6.

net


10. p  m v

 
dp
11.  F  Fnet 
dt



J

F
dt


p
12.



13. P   p i


dP
14.
  Fext
dt
15. M   m i
16.
17.
1
x cm   m i x i
M


P  M v cm
y cm 
1
 mi yi
M
R2-43
Formula Sheet Organization
Work and Energy (Linear Motion)
   
18. a  b  a b cos( )  a x b x  a y b y
 
19. W  F  d

W

F

d
x
20.

23.
K  12 m v 2  12 m ( v x  v y )
K f  K i  Wnet


U    Fcons  dx
24.
U g  m g (y  y 0 )
25.
U s  12 k ( x  x 0 ) 2
 K   U  Wnon cons
21.
22.
26.
2
2
R2-44
Formula Sheet Organization
Rotational Kinematics
30.
   0   t  t 0 
31.
   0   0 ( t  t 0 )  12  ( t  t 0 ) 2
32.
   0  12 ( 0  )( t  t 0 )
33.
   0  ( t  t 0 )  12  ( t  t 0 ) 2
34.
 2  02  2   0 
R2-45
Formula Sheet Organization
Rotational Motion / Linear Motion
7.
2r
T
v
8.
a centripetal
9.
Fcentripetal
27.
28.
29.
s  r
v2

 2 r
r
v2
m
 m 2 r
r
v tan gential   r
a tan gential   r
R2-46
Formula Sheet Organization
Newton’s 2nd Law and Angular Mom.
   
35. a  b  a b sin( )
36.
39.
40.
41.
42.
43.
I   m i ri
  
  rF
2

 dL

  I  d t
  
l  rp


L   li


L  I
R2-47
Formula Sheet Organization
Work and Energy (Rotational)
37.
38.
K rot  12 I 2
 
W     d
R2-48
Formula Sheet Organization
Collisions
44x. m 1 v1, x ,before  m 2 v 2, x ,before  m 1 v1, x ,after  m 2 v 2, x ,after
44y. m1 v1, y ,before  m 2 v 2, y ,before  m 1 v1, y ,after  m 2 v 2, y ,after
m1  m 2
2 m2
v1,i 
v 2 ,i
45a. v1,f 
m1  m 2
m1  m 2
2 m1
m 2  m1
v1,i 
v 2 ,i
45b. v 2,f 
m1  m 2
m1  m 2
R2-49