Physics 218
Review
Prof. Rupak
Mahapatra
Physics 218, Chapter 3 and 4
1
Checklist for Final
•Work out all past finals from webpage
•Do ALL end of chapter exercises from all
chapters
–The final questions are typically text book style
questions
•Look up your final schedule
Physics 218, Chapter 3 and 4
2
Projectile Motion
The physics of the universe:
The horizontal and
vertical Equations of
Motion behave
independently
This is why we use vectors in the
first place
Physics 218, Chapter 3 and 4
3
How to Solve Problems
The trick for all
these problems is to
break them up into
the X and Y
directions
Physics 218, Chapter 3 and 4
4
Firing up in the air at an angle
A ball is fired up in the air with speed Vo and
angle Qo. Ignore air friction. The acceleration
due to gravity is g pointing down.
What is the final velocity here?
Physics 218, Chapter 3 and 4
5
Maximize Range Again
• Find the minimum initial speed of a
champagne cork that travels a
horizontal distance of 11 meters.
Physics 218, Chapter 3 and 4
6
Translate: Newton’s Second Law
The acceleration is in
the SAME direction
as the NET FORCE
 This is a VECTOR
equation
 If I have a force,
what is my
acceleration?
 More force → more
acceleration
 More mass → less
acceleration
Vector Equation :


F  ma
Fx  ma x , Fy  ma y


Weight  W  mg
Physics 218, Chater 5 & 6
7
Pulling a box
A box with mass m is pulled along a frictionless
horizontal surface with a force FP at angle Q as
given in the figure. Assume it does not leave the
surface.
a)What is the acceleration of the box?
FP Q
b)What is the normal force?
Physics 218, Chater 5 & 6
8
2 boxes connected with a string
Two boxes with masses m1 and m2 are placed on a
frictionless horizontal surface and pulled with a
Force FP. Assume the string between doesn’t
stretch and is massless.
a)What is the acceleration of the boxes?
b)What is the tension of the strings between the
boxes?
M2
M1
Physics 218, Chater 5 & 6
9
The weight of a box
A box with mass m is resting on a
smooth (frictionless) horizontal
table.
a) What is the normal force on the
box?
b) Push down on it with a force of FP.
Now, what is the normal force?
c) Pull up on it with a force of FP such
that it is still sitting on the table.
What is the normal force?
d) Pull up on it with a force such that
it leaves the table and starts rising.
What is the normal force?
Physics 218, Chater 5 & 6
FP
10
Atwood Machine
Two boxes with masses m1
and m2 are placed around
a pulley with m1 >m2
a) What is the acceleration
of the boxes?
b) What is the tension of
the strings between the
boxes?
Ignore the mass of the
pulley, rope and any
friction. Assume the rope
doesn’t stretch.
Physics 218, Chater 5 & 6
11
Kinetic Friction
• For kinetic friction, it turns out that
the larger the Normal Force the
larger the friction. We can write
FFriction = mKineticFNormal
• Warning:
Here m is a constant
– THIS IS NOT A VECTOR
EQUATION!
Physics 218, Chater 5 & 6
12
Static Friction
• This is more complicated
• For static friction, the friction force
can vary
FFriction  mStaticFNormal
Example of the refrigerator:
– If I don’t push, what is the static
friction force?
– What if I push a little?
Physics 218, Chater 5 & 6
13
Two Boxes and a Pulley
You hold two boxes, m1 and
Ignore the mass of the pulley
m2, connected by a rope
running over a pulley at
and rope and any friction
rest. The coefficient of
associated with the pulley
kinetic friction between
the table and box I is m.
You then let go and the
mass m2 is so large that
the system accelerates
Q: What is the magnitude of
the acceleration of the
system?
Physics 218, Lecture IX
14
An Incline, a Pulley and two Boxes
In the
diagram
given, m1 and
m2 remain at
rest and the m2
angle Q is
known. The
coefficient
of static
Ignore the mass of the pulley
and cord and any friction
associated with the pulley
Physics 218, Lecture IX
Q
15
Skiing
You are the ski designer
for the Olympic ski team.
Your best skier has mass
m. She plans to go down a
mountain of angle Q and
needs an acceleration a in
order to win the race
What coefficient of
friction, m, do her skis
need to have?
Q
Physics 218, Lecture IX
16
Is it better to push or pull?
You can pull or push a sled with the same force magnitude,
FP, but different angles Q, as shown in the figures.
Assuming the sled doesn’t leave the ground and has a
constant coefficient of friction, m, which is better?
FP
Physics 218, Chater 5 & 6
17
Work for Constant Forces
The Math: Work can be complicated.
Start with a simple case
Do it differently than the book
For constant forces, the work is:
.
W=F d
…(more on this later)
Physics 218, Chapter 7 & 8
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Find the work: Calculus
To find the total work, we must sum up all the little
pieces of work (i.e., F.d). If the force is continually
changing, then we have to take smaller and smaller
lengths to add. In the limit, this sum becomes an integral.
b

a F

 dx
Total sum
Integral
Physics 218, Chapter 7 & 8
20
Non-Constant Force: Springs
• Springs are a good
example of the types of
problems we come back
to over and over again!
• Hooke’s Law


F  kx
Some constant
Displacement
• Force is NOT
CONSTANT over
a
Physics 218, Chapter 7 & 8
distance
21
Work done to stretch a Spring
How much work
do you do to
stretch a spring
(spring constant
k), at constant
velocity (pulled
slowly), from
x=0 to x=D?
Physics 218, Chapter 7 & 8
D
22
Work Energy Relationship
• If net positive work is done on a
stationary box it speeds up. It now has
energy
• Work Equation naturally
leads to derivation of kinetic
energy
Kinetic Energy = ½mV2
Physics 218, Chapter 7 & 8
23
Work-Energy Relationship
•If net work has been done on an
object, then it has a change in its
kinetic energy (usually this means
that the speed changes)
•Equivalent statement: If there is a
change in kinetic energy then there
has been net work on an object
Can use the change in energy to
calculate the work
Physics 218, Chapter 7 & 8
24
Summary of equations
Kinetic Energy =
2
½mV
W= DKE
Can use change in speed
to calculate the work, or
the work to calculate the
speed
Physics 218, Chapter 7 & 8
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Conservation of Mechanical Energy
• For some types of problems,
Mechanical Energy is conserved (more
on this next week)
• E.g. Mechanical energy before you
drop a brick is equal to the mechanical
energy after you drop the brick
K2+U2 = K1+U1
Conservation of Mechanical Energy
E =E
Physics 218,
2 Chapter
1 7&8
26
Problem Solving
• What are the types of examples we’ll
encounter?
– Gravity
– Things falling
– Springs
• Converting their potential energy into
kinetic energy and back again
E = K + U =
2
½mv
Physics 218, Chapter 7 & 8
+ mgy
27
Problem Solving
For Conservation of Energy
problems:
BEFORE and AFTER
diagrams
Physics 218, Chapter 7 & 8
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Potential Energy
A brick held 6 feet in the air has
potential energy
• Subtlety: Gravitational potential
energy is relative to somewhere!
Example: What is the potential energy of a book 6
feet above a 4 foot high table? 10 feet above
the floor?
• DU = U2-U1 = Wext = mg (h2-h1)
• Write U = mgh
• U=mgh + Const
Only change in potential energy is really
Physics 218, Chapter 7 & 8
29
meaningful
Other Potential Energies: Springs
Last week we
calculated that it
2
took ½kx of work
to compress a
spring by a distance
x
How much potential
2
U(x) does
= ½kx
energy
it now
how have?
Physics 218, Chapter 7 & 8
30
Energy Summary
If work is done by a non-conservative force
it does negative work (slows something
down), and we get heat, light, sound etc.
EHeat+Light+Sound.. = -WNC
If work is done by a non-conservative force,
take this into account in the total energy.
(Friction causes mechanical energy to be
lost)
K1+U1 = K2+U2+EHeat…
K1+U1 = K2+U2-WNC
Physics 218, Lecture XII
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Force and Potential Energy
If we know the potential energy, U, we
can find the force
Fx  
dU
dx
This makes sense… For example, the
force of gravity points down, but the
potential increases as you go up
Physics 218, Lecture XIII
32
Mechanical Energy
• We define the total
mechanical energy in a
system to be the kinetic
energy plus the potential
energy
• Define E≡K+U
Physics 218, Lecture XIII
33
Conservation of Mechanical Energy
• For some types of problems, Mechanical
Energy is conserved (more on this next
week)
• E.g. Mechanical energy before you drop a
brick is equal to the mechanical energy
after you drop the brick
K2+U2 = K1+U1
Conservation of Mechanical Energy
E2=E1
Physics 218, Lecture XIII
34
Friction and Springs
A block of mass
m is traveling on
a rough surface.
It reaches a
spring (spring
constant k) with
speed Vo and
compresses it a
total distance D.
Physics 218, Lecture XV
35
Robot Arm
A robot arm has a funny
Force equation in 1dimension
2

3x
FX  F0  1  2
x
0





where F0 and X0 are
constants. The robot
picks up a block at X=0
(at rest) and throws it,
releasing it at X=X0.
What is the speed of
the block?
Physics 218, Lecture XV
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Overview: Rotational Motion
– Position
– Velocity
– Acceleration
– Force
– Mass
– Momentum
– Energy
←
←
←
Physics 218, Chapter 12
Start
here!
Chapters 1-3
• Take our results from “linear” physics
and do the same for “angular” physics
• Analogue of
49
Velocity and Acceleration
Define  as the angular velocity
D
d
 
or  
radians/ sec
Dt
dt
Define  as the angular acceleration
d
d 
 
or  
2
dt
dt
2
radians/ sec
Physics 218, Chapter 12
50
2
Right-Hand Rule
Yes!
Define the
direction to
point along
the axis of
rotation
Right-hand
Rule
This is true
for Q,  and

Physics 218, Chapter 12
51
Uniform Angular Acceleration
Derive the angular equations of motion
for constant angular acceleration
1 2
Q  Q0 0t  t
2
  0  t
Physics 218, Chapter 12
52
Rolling without Slipping
• In reality, car tires both
rotate and translate
• They are a good example of
something which rolls
(translates, moves forward,
rotates) without slipping
• Is there friction? What kind?
Physics 218, Chapter 12
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Derivation
• The trick is to pick your
reference frame
correctly!
• Think of the wheel as
sitting still and the
ground moving past it with
speed V.
Velocity of ground (in bike
frame) = -R
=> Velocity of bike
(in
Physics 218, Chapter 12
ground frame) = R
54
Centripetal Acceleration
• “Center Seeking”
• Acceleration
vector= V2/R
towards the
center
v
a 
( ˆ
r)
R

2
ˆ
r direction
• Acceleration is
perpendicular to
the velocity
Physics 218, Chapter 12
R
55
Circular Motion: Get the speed!
Speed = distance/time
Distance in 1 revolution divided by
the time it takes to go around
once
Speed = 2pr/T
Note: The time to go around once is
known as the Period, or T
Physics 218, Chapter 12
56
More definitions
• Frequency = Revolutions/sec
 radians/sec
/2p
f = 
• Period = 1/freq = 1/f
Physics 218, Chapter 12
57
Ball on a String
A ball at the end of a string is
revolving uniformly in a
horizontal circle (ignore
gravity) of radius R. The ball
makes N revolutions in a time
t.
What is the centripetal
acceleration?
Physics 218, Chapter 12
58
The Trick To Solving Problems

F

 ma
v
 m 
R
2

( ˆ
r)

Physics 218, Chapter 12
59
Banking Angle
You are a driver on
the NASCAR circuit.
Your car has m and
is traveling with a
speed V around a
curve with Radius R
What angle, Q,
should the road be
banked so that no
friction is required?
Physics 218, Chapter 12
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