Transcript PHY 101 Lecture Notes

PHY 101
Lecture Notes
Instructor: Laura Fellman
Chapter 2
A brief look at the historical
development of physics and
Newton’s 1st Law of Motion
Aristotle (384-322 BC)

Greek philosopher/scientist
 Aristotle was an observer not an experimenter
– He thought there were 2 classes of motion:
(1) natural motion:
every object in universe has a proper place
and strives to get to this place
(2) violent motion = imposed motion
results from pushing and pulling forces
WE NOW KNOW ARISTOTLE WAS WRONG!
Nicolaus Copernicus (1473-1543)
• Polish astronomer who changed astronomy profoundly
• 1510: derived a heliocentric or “sun-centered” model
• Only published in 1543: “De Revolutionibus”
• Book was banned by Church between 1610 & 1835
• Now we recognize Copernicus as a “giant” in
astronomy
Galileo Galilei (1561 – 1642)

professor of Mathematics at University in Italy
 Galileo used observations and
experiments to disprove Aristotle’s
ideas
 he was interested in HOW things
moved, not why they moved.
 we call this kinematics
 Important experiment: Galileo dropped heavy
and light objects together and found they hit the
ground at the same time.
– See the experiment in action
Air resistance

Air resistance affects motion and makes it more
complicated
– See Elephants and feathers

If we can ignore air resistance, we find that the
relationships describing motion are simpler
 When can we neglect air resistance?
(1) If there is no air! (in a vacuum)
(2) If the objects in motion are:



heavy
compact (dense)
traveling at moderate speeds
Back to Galileo

Galileo stated that:
If there is no interference with a moving object,
it will keep moving in a straight line forever.

See Web demo example
 Consider an experiment in which you:
– roll a ball up an incline
– roll a ball down an incline
– along a flat surface

see Figure 2.3 in text and an online explanation
Galileo & the telescope
• In 1608 a Dutch lens maker invented the telescope
• Galileo built one in 1609
• In 1610 he published “The Starry Messenger”
documenting many important observations,
including
– Moon’s surface had features (mountains & valleys)
– Milky Way was made up of many stars
– Jupiter had moons circling it
• Soon after this he also discovered:
– Sun was not perfect but had “spots” on its surface
– Sun was spherical & rotated about its own axis
– Venus went through complete set of phases like Moon
Galileo in trouble
• In 1632 Galileo publishes “Dialogue Concerning the
Two Chief World Systems” defending Copernicus
• Interrogated by the Inquisition
• In 1633 he recants and admits his errors
• Sentenced to life house arrest where he dies
• In 1992 Catholic church finally officially admits that
Galileo was right
Newton(1642-1727)
Changed the focus from “how” to “why”
 Made brilliant contributions to physics!
 Pondered why apple fell to
Earth amongst other things
 He summarized his findings
in 3 laws = Newton’s Laws
 All involve the idea of a force
(or lack of a force)

Isaac Newton: Yes, the apple really fell!
•
Published “Principia” in which he outlined 3
basic laws of motion:
1. A body continues at rest or in motion in a
straight line unless acted on by some force.
2. The change in motion of a body is
proportional to the size and direction of the
force acting on it.
3. When one body exerts a force on a 2nd
body, the 2nd body exerts an equal &
oppositely directed force on the first.
Newton’s First Law/ Law of Inertia
An object at rest remains at rest if no force acts on it
An object in motion remains in motion if no force acts on
it
 Inertia = resistance of an object to
a change in its motion
 See this in action
 Experience tells us that the heavier
an object is, the harder it is to get it
up to speed when pushing it.
 Scientifically we could say: the greater the object’s
mass, the greater its resistance to a change in its
motion.
 So mass is a measure of an object’s inertia.
Force

Can think of force as a push or pull action
 What causes this push or pull?
– Contact force
– Electrical force
– Magnetic force
non-contact force
– Gravitational force

Forces result in a change of motion

What if more than one force acts at a time?
Net force

Need to combine the forces & find net force
3N
2N
Fnet ?
2N
2N
4N
3N
2N
Fnet ?
Fnet ?
Review of Law of Inertia
See this online summary
Equilibrium

Condition for equilibrium: Fnet = 0
– so all forces balance each other

Static equilibrium:
speed = 0 (no motion), and
Fnet = 0

Support forces
Q. What stops a book from falling through the
table it lies on?
Ans: A support or “Normal” force

What’s normal about it?
Examples:

How does a scale work?
– Identify what forces are involved
– what is the sum of these forces?
– Spring stretches (compresses) by an amount proportional
to force that pulls (pushes) on it
– See this in action

Standing on one scale:
– What is the net force?

Now stand on 2 scales:
– what does each scale read?
– How would scale readings change if
you shift your weight?
Tension

Tension (T) is a type of force (like gravitational
force or electric force are force types)
 It is a “pulling” force usually exerted on an object by
a rope or a chain
 Pulleys: change direction of force, not the magnitude
2
1
3
T1 , T2 and T3 are all
equal in size, but in
different directions.
Examples:

Window washers: Joe and Jane
(equal weights)





spring scale
T1 = ?
T2 = ?
What are T1 and T2 ?
What if Jane, on right, walks over towards Joe?
What happens to T1 and T2 now ?
What happens to the total tension (T1 + T2 )
How are T1 and T2 related to each other?
spring scale
T1 = ?
T2 = ?

Let’s try practice pages 3 and 4 now in your
Practicing Physics book

Then we’ll try this question……
Dynamic equilibrium

Conditions for “moving” equilibrium:
– Still need net force on object = 0
– object moves at constant velocity

Example:
– Flying at constant speed in airplane
 Key is you can’t feel that you are moving

When do we get a sensation of motion?
Chap 3: Linear Motion
Let’s find ways to describe how
things move
Description of Motion

We will consider motion in terms of:
 distance, and
 time

Graphs are a great way to visualize motion.

First consider only position or distance from a point:
0
1
2
3
4
x-axis in meters

object starts at zero marker and moves, in 1 meter steps, to the
3 meter mark

Now we include time
– record where the object is and when it gets there

As before we can graph our position but now in
relation to time
position (x)
in [m]
4
3
2
1
0
1
2
3
4
time (t) in [seconds, s]
See motion being graphed in passing lane demo
Distance and time

We can combine distance and time knowledge to
get the following quantities:
– Speed: how fast?
– Velocity (v): how fast and in what direction?
– Acceleration (a): how quickly does v change?
Speed: how fast?
distance
speed =
time

Units: km/hour or

Two ways to look at speed:
(1) average speed
(2) instantaneous speed
mph
or
m/s
SI Unit for speed
Average speed
Objects don’t always travel at same speed
 Example: driving your car

– drive to Seattle (180 miles) in 3 hours
– may stop, get stuck in traffic, etc

Can still determine my average speed:
average speed =
total distance covered
time interval
Instantaneous speed

Speed at any one instant
 Example: when driving your speed changes
– instantaneous speed = speed on your speedometer
Special case: if your speed is constant for whole journey,
then: instantaneous speed at all times = average speed
Graphing speed vs time

Just like we graphed position vs time, we can graph velocity as
it changes with time.
position (x)
in [m]
4
3
2
1
velocity (v) 4
3
in [m/s]
2
1
0

1 2 3 4 time (t)
0
1
2
3 4 time (t)
Let’s go back to the passing lane demo and graph v vs time
now instead of x vs time.
Examples involving distance and speed

Let’s try some conceptual questions:
– Motorist
– Bikes and Bees
More on average speed

A reconnaissance plane flies 600 km away
from its base at 200 km/h, then it flies back
to its base at 300 km/h.
What is the plane’s average speed?
Velocity

Now we consider speed and direction
 Example:
– speed = 50 km/h
– velocity = 50 km/h to the south

constant speed: equal distances covered in equal time
intervals
 constant velocity = constant speed and no change in
direction
 Ex 1: car moves around a circular track
– constant speed
– but velocity not constant!
Speed vs Velocity

Here is example where average speed and
average velocity are very different.
 Example: Walking the dog
– The owner and the dog have the same change in
position but the dog covers much more distance
in the same time, so they have the same average
velocity but very different average speeds.
– See also a similar online demo of this idea
Acceleration

Acceleration

acceleration: speeding up or slowing down
=
=
rate of change of velocity
change of velocity
time interval
Q. Can we feel velocity?
Q. Can we feel acceleration?
Q. What controls in a car make it accelerate?
Examples
Ex 1: A car starts at rest and reaches 60 mi/hr in 10s.
Q. What is the car’s acceleration?
Acceleration = (change in v) = 60 mi/hr = 6 mi/hr.s
time
10 s
Ex 2: A cyclist’s speed increases from 4 m/s to 10 m/s
3 seconds.
Q. What is the cyclist’s acceleration?
in
Graphs showing acceleration

What does a velocity vs time graph look like when an
object is accelerating?

Let’s go back to our car demo and see what this looks
like in the stoplight scenario

Now lets look at 3 graphs of the same motion:
1.
2.
3.
position vs time
velocity vs time
acceleration vs time
Acceleration on inclined planes
Q. On which of these hills does the ball roll down
with increasing speed and decreasing acceleration
along the path?
A
B
(Hint: see Fig 3.6 in textbook)
C
Free fall

Things fall due to the force of gravity
 if there are no restraints (air resistance) on object,
we say the object is in FREE FALL
 acceleration due to gravity is approximately
g = 10 m / s2
(meters per second squared)

The actual value is closer to g = 9.8 m / s2
When objects fall, we will ask……..
 How fast?
 How far?

How fast and how far?

Q. If an object is dropped from rest (no
initial velocity) at the top of a cliff, how fast
will it be travelling:
– after 1 second?
– after 2 seconds?

Q. How far does object drop in 1s?

Why?
Summary: Motion relationships

Instantaneous velocity for an object that starts at rest:
v = acceleration * time
(in general)
= gravity * time
(for free fall object)
or for an object that starts with an initial speed
v = initial velocity + a * t
= initial velocity – g * t
(up is positive)




Distance traveled for an object that starts at rest:
d = ½ acceleration * (time)2 (general)
=½g*t2
(for free fall)
Distance traveled for an object that starts with an initial speed
d = initial velocity * time + ½ acceleration * (time)2
= initial velocity * t - 1/2 g t2
Remember to use correct units: if g has units of m / s2 then you
must use time in seconds.
Examples

Look over Practice pages 5 and 6

Example:
A ball is dropped from rest from a height of
20m. How long does it take to reach the
ground?
Chapter 4: Newton’s 2nd Law
Why things move
Newton’s 2nd Law of Motion
• The acceleration (a) of an object is:
– directly proportional to the net force (Fnet) acting
on it,
and
– inversely proportional to the mass (m) of the
object
• In symbols we can write:
a = Fnet / m
• NOTE: acceleration and force both have a direction
and a magnitude associated with them
– direction of “a” is given by the direction of Fnet
Notation:
N
Normal force
(contact force)
F
Pulling or pushing force
W
Weight
(gravitational force)
Example: If the block has a mass of 10 kg and if pulled by
a force of 50N, find the values of the forces shown in the
above diagram and calculate the horizontal acceleration.
Rank the accelerations, smallest to largest
A
B
D
C
Mass, Weight & Volume
• Mass: how much “stuff” something is made of
– measure of an object’s inertia:
more mass = more inertia
– UNITS of measurement: [kg] or [grams]
• Weight: force on an object due to gravity
– UNITS of measurement: [Newton, N] (metric unit)
or [pounds, lbs]
• Volume: mass is not volume!
– Massive doesn’t mean voluminous
– something can be massive (heavy) but not large
– this object has a high density = (mass) / (volume)
Examples


What are the mass and weight of a 10 kg block on:
(a) the Earth
(b) moon
A 50 kg woman in an elevator is accelerating upward at
a rate of 1.2 m/s2.
(a) What is the net force acting on the woman?
(b) What is the gravitational force acting on her?
(c) What is the normal force pushing upward on
the woman’s feet?
See a demo of an elevator ride in action
Newton’s 2nd Law in many object problems
Let’s try an example where there are several objects involved:
Three blocks of equal mass (2kg) are tied together. If you
pull on one end with a force of 30N, what are the tensions in
the other two ropes that join the blocks together?
T2 = ?
T3 = ?
2kg
2kg
2kg
T1 = 30N
Friction
• Now we are ready to start considering the
effects of friction
• drag a block across surface
– know there is friction between surface and block
– if speed of the block, v = constant, then
a=0
– so by Newton’s 2nd Law:
Fnet = 0
Now we have Dynamic Equilibrium
Conditions: v = constant & a = 0
Friction
Need: 2 surfaces are in mutual contact
– magnitude of frictional force?
• depends on the type of surfaces in contact
Which is harder to push?
• depends on the weight of the object
– direction of frictional force?
Direction of motion
• in opposite direction to motion
• What causes the friction?
– Irregularities (roughness) in surfaces
Frictional force
2 kinds of friction
• Static friction: before there is any motion
v=0
Friction = 70N
Applied force = 70N
• Sliding friction: when block is motion
v = constant
Friction = 50N
Applied force = 50N
• Static friction > sliding friction
Interesting facts about friction
• Does not depend on: speed and contact area
So then:
• Why do trucks have so many tires?
• Why do high performance cars have wide tires?
Force at angles & friction
support force
tension
friction
vertical
part of
tension
horizontal part of tension
weight
Free-fall revisited
• Let’s ignore air drag just for a moment:
• A heavy object experiences a larger gravitational
force so you might think that it has a larger
acceleration than a light object (a ~ F)
BUT
• The heavy object has greater inertia & so it has a
greater resistance to change in it motion so you might
think it has a smaller acceleration than a light object
(a ~ 1/m)
• Actually: combine both of these and get that the two
objects have the same acceleration!
a ~ Fnet / m
(see Fig 4.10)
Friction in fluids (drag)
• Fluids = things that flow
– gas (e.g. air) or liquid (e.g. water)
• What does drag, or resistance in fluids,
depend on?
– properties of fluid (density)
– speed (in lab we will determine the exact
relationship)
– area of contact
• So friction in fluids is very different to friction
between 2 solids in contact!
Non-Free fall
• 2 equal masses are dropped, but have different
surface areas.
– Which hits the ground first?
• 2 parachuters (one heavy, one light) jump out of
an airplane.
– Which one of the two falls faster?
• What is going on?
• free fall: only gravitational force (weight)
– so net force Fnet = W
• non-free fall: Must now also consider the air
resistance (R)
– now net force Fnet = W - R
Terminal velocity
• Terminal velocity is achieved when falling object
is no longer accelerating (a = 0, v = max)
– since Fnet = W - R
– Acceleration: a = Fnet / m = (W - R) / m
– so a = 0 when W = R
• Recall that R depends on speed, so as speed
increases R increases until eventually R = W
– If W is small then R = W sooner (at a lower velocity)
then for large W
– So if 2 objects are the same size, the heavier one will
have a greater terminal velocity
Let’s consider some examples
First let’s look at the force of air resistance
 Let’s revisit the elephant and the feather
falling
 Now let’s see what happens during a
skydiver’s journey to the ground

– First you try practice page 10
– Then we’ll look at a demo to see the jump and
forces in action:
– Animated skydiver
Chapter 5: Newton’s 3rd Law
and
Vectors
Newton’s 3rd Law
Whenever object A exerts a force on object B,
object B exerts an equal and opposite force
on object A.
• refer to these as action & reaction forces
– see Fig 5.5
– Hand pushes on table (action)
– Table pushes on hand (reaction)
• How to identify these force “pairs”:
• always involves 2 forces
• the forces are acting on different objects
Examples and Problems
First we’ll consider the question of an apple
on a table.
 Now look at these examples of force pairs
 And try the tutorial at your textbook website


Example: A horse pulls a cart. If the cart
exerts a force on the horse that is equal and
opposite to the force that the horse exerts on
the cart, why does the cart move?
– Again see the textbook website tutorial for more
Defining your system

So do action/reaction forces cancel each other?
– No! Careful: they are not acting on the same body!
 Example: apple pulls on orange in a cart
– Consider 3 different systems:
1. The orange only
2. The apple only
3. Orange & apple together
Frictional force
= external force
Combining Newton’s 2nd and 3rd laws

Example 1:
B
A
150 N
5 kg
10 kg
Ice, so no friction to worry about
Find: (a) acceleration of the blocks
(b) force on block B by block A
(c) force on block A by block B

Example 2: A 56 kg parent and a 14 kg
child are ice skating. They face each other
and push on each other’s hands.
(a) Which person experiences a bigger force?
(b) Is the acceleration of the child larger, the
same, or smaller than the parent’s
acceleration?
(c) If the acceleration of the child is 2.6 m/s2,
what is the parent’s acceleration?
2D motion and Vectors
2-D Motion



Till now:
– 1-D motion: motion along a line
– position, speed, velocity and acceleration
Now:
– 2-D motion
– Motion in the horizontal & vertical directions or in a circle
– will need a new way to represent this motion
Several topics related to 2-D motion
– circular motion (return to this later)
– relative motion
covered at end of Chap 5
– vectors
– projectile motion (beginning of Chap 10)
Relative Motion

tailwind
tailwind
headwind
headwind
Airplane flies:
– faster with a tailwind
– slower into a headwind

same is true when you ride your bike!
 What happens in a crosswind?
now have 2-D motion
 Need to introduce vectors
Vectors

Imagine the following:
–
–
–
–

you’re riding on the bus with a physicist
you decide to ask her how things work
all the physicist has on hand is an envelope
What happens?
Vectors: arrows that illustrate both:
– size
– Direction
– Examples:

Scalars:
– only size
– Examples:
Vector example (1-D)
wind

Consider the airplane:
– airplane’s velocity: vA = 100 km/h to north
– tailwind: vw = 20 km/h to north
– What is the plane’s speed relative to
the ground?
vR = vA + vw = 100 km/h + 20 km/h
= 120 km/h
plane
resultant
wind
– Now consider a headwind:
vw = 20 km/h to south
vR = vA + vw = 100 km/h + (-20 km/h)
= 80 km/h

plane
resultant
Crosswind (2-D)
– airplane’s velocity: vA = 80 km/h to north
– crosswind: vw = 60 km/h to east


Want to add 2 vectors and get the resultant vector
Use parallelogram method:
 complete “box” by adding parallel lines
 draw a diagonal from the starting point of 2 vectors
– To find the length of the diagonal (resultant):


scale drawing (measure)
Pythagorean Theorem: c2 = a2 + b2
or
c = a2 + b2
– To find the direction of the resultant:


scale drawing (measure with a protractor)
use trigonometry
Chapter 10: Projectile Motion
a.k.a. How to hit your neighbors with a cannon ball!
?
Projectile motion
• When an object is given:
– an initial horizontal velocity
– experiences the force of gravity (vertical direction)
we call the object a projectile
the path the object follows is its trajectory
• How do we determine the projectiles trajectory?
• We note that:
THE VERTICAL AND HORIZONTAL MOTION OF AN
OBJECT DO NOT AFFECT EACH OTHER!
• Dropping ball demo
Target practice: Horizontal Launch
• Remember: we can consider the horizontal
component of motion and the vertical
component separately!
• If I aim directly at the target and it takes my
arrow 1 second to reach the target, where
does my arrow end up?
• Horizontal projectile launch
?
If you say it hits below,
how far below the target?
?
Now add in the
velocity components
Firing at an angle
• Let’s consider what happens when a Zookeeper fires
a banana at a monkey
• And then let’s see this in action
• Now consider Fig 10.6 in your textbook
• The cannon now fires upward at some angle
• How do we figure out its trajectory?
– First, consider what path projectile will follow if
gravity was not present
(in other words, a straight line)
– Then after each second, consider how far
projectile would fall straight down
• We see that the trajectory of the projectile has the
mathematical shape known as a parabola
Velocity of a projectile
•
Let’s take a look at a demo of cannonball
that is fired at an angle.
• The horizontal and vertical parts (components)
of the ball’s velocity are shown
• We can combine these two components by
adding them as vectors using the parallelogram
method to give us the velocity of the ball at any
point.
•
Important: no acceleration in horizontal
direction so projectile moves equal
horizontal distances in equal time intervals.
How high & how far?
• What do we know when we fire the cannonball?
– its launch angle
– its launch (muzzle) speed
• Then we might want to know:
– How high does it go? = vertical part of the problem
– How far does it go? = horizontal part of the problem
• KEY POINT: Since the horizontal & vertical
motions don’t affect each other we can treat
them separately.
• How high?
– need the vertical component of the launch velocity
– then we can solve it as if we threw the ball straight up
at that vertical speed (as we did in Chap 3)
– figure out how long (time) before the ball comes to a
stop, in other words, when is vvertical = 0?
– then the distance it goes up is:
distance up = distance down = y = ½ g t2 = 5 t2
•
How far? (call this the range of the projectile)
– need the horizontal component of the launch velocity
– need to know how long (time) the ball stays in the air (see
the how high section to get time)
– Then since there is no acceleration in the horizontal
direction (gravity is in the vertical only) we get that:
Range = dacross = x = vhorizontal * t

Let’s test our understanding with a battleships question
Launch angle
• How does launch angle affect the range?
• Let’s take a trip to the golf range to test things
• Experiment: Keep the launch speed the same and
change the angle to observe the effect.
• Findings:
– Maximum range achieved at 45o (no air resistance)
– also, complimentary angles give same range:
• 15o and (90o - 15o ) = 75o
• 30o and (90o - 30o ) = 60o , etc
• NOTE: air resistance is important, especially for fast moving objects
(like baseballs)
– max range not at 45o when air resistance is taken into account
(more about this in the Lab this week)
Example 1
• A football player throws a football level to the
ground from a height of 1.5 meters. The ball
lands 20 meters away from him. How fast
was the football going when it left the player’s
hand?
1.5 m
20 m
Example 2

A red cross airplane flying level at a speed of 40
m/s must drop relief supplies. If the plane is flying
at a height of 500m, how far before the landing
site must the plane drop the package?
40 m/s
500 m
?
Example 3
A cannon is fired over level ground at an angle of 30
degrees to the horizontal. The initial velocity of the
cannonball is 200 m/s. That means the vertical
component of the initial velocity is 100 m/s and the
horizontal component is 173 m/s.
(a) How long is the cannonball in the air for?
(b) How far does the cannonball travel horizontally?
(c) Repeat the problem but with a launch angle of 60
degrees. This means the vertical component of the
initial velocity is now 173 m/s and the horizontal
component is 100 m/s.
Example 4
• Cannonball fired:
muzzle speed = 141 m/s
launch angle = 45o
• It hits a balloon at top of its trajectory.
• What is the velocity of the cannonball when it
hits the balloon? (Neglect air resistance)
Chapter 6: Momentum
Chapter 6 : Momentum


Momentum
is
inertia (m) in motion (v)
momentum
=
mass * velocity
p
=
m * v
UNITS: kg m /s
(no special name)
Values of momentum
m = 7 kg
v = 2 m/s
p = 14 kg m /s
m = 0.070 kg
v = 200 m/s
p = 14 kg m /s

We can get large momentums when:
– mass is large
(supertanker, p = Mv)
– velocity is large (major league fastball, p = mV)
– both these are large
(Boeing 747, p = MV)
Force & Momentum

How do we change momentum?
– change mass, change velocity or change both
momentum =
m * v
usually keep this same
change v
So we have acceleration
have a net force acting
When there is an external force on system, then momentum changes
How force changes momentum
F/m = a

(Newton’s 2nd Law)
now multiply both sides by t and m
t * m * F = a * t * m = change in v * t * m
m
t
 this leaves us with:
Ft = change in (mv)
Impulse = change in momentum

Racquetball hitting the wall
Changing momentum

We can consider various changes in momentum and the
impulse that produces this change:
– Increasing momentum
– decreasing momentum over a long time
– decreasing momentum over a short time
Increasing momentum:
F
When will final velocity be greater:
short push or long push?
Decreasing momentum over a long time:
 Truck moves with velocity, v. When your brakes fail
and you want to stop (v=0) do you:
– slam into a haystack?
– slam into a concrete wall?
Hint: the change in momentum is same in both cases
want to try to minimize the force you feel.
Decreasing momentum over a short time:
 now goal is to maximize force
 Ex: break a stack of bricks with your hands.

Let’s look at a web demo of a car slowing down
Some more examples
1. Which has more momentum: a truck at rest
or a dragonfly flying over a pond?
?
?
2. A car with a mass of 1000 kg moves at
20m/s. What braking force is needed to
bring it to a stop in 10s?
Conservation of momentum


When a physical quantity remains unchanged during a process, we
say that the quantity is conserved
So “Conservation of momentum” means that momentum remains
unchanged, or
Momentum before = momentum after

When is momentum conserved?
Momentum of a system is conserved when no
external forces act on that system
Web demo: momentum cart
Example

Rifle fires a bullet or a cannon fires a cannonball
– Web demo: cannonball fired
m
1. Draw situation before
the action (firing of rifle):
M
Draw the situation after the firing of the rifle
3. Identify a system on which there are no external
forces acting
4. For this system the momentum is conserved:
2.
(momentum of system)before = (momentum of system)after
More Examples

Let’s try an example where a girl jumps off
a heavy, stationary cart. As we can guess by
now, the cart will move in the opposite
direction to the girl and we can figure out
how fast if we know a few things. So let’s
look at the web demo of the girl jumping off
a cart and do some calculations.

We can see this same principle at work
when a rocket ejects a pellet for propulsion
Now you try one:
Two ice skaters are standing still in the middle of
the ice when they push off each other. The one
skater has a mass of 100 kg, while the other has a
mass of 50 kg. If the 100 kg skater has a speed of
2 m/s, what is the speed of the lighter skater?

Here’s some questions you should ask yourself as you
work through this:
– Are there any external forces acting?
– Do you expect the smaller skater to be moving faster or
slower than the large skater?
– Which directions do the skaters move in?
Collisions

Conservation of momentum is also useful for
solving problems involving collisions

Elastic collisions (web demo)
– Colliding objects rebound
– no deformation of the objects involved

Inelastic collisions
– objects become entangled
– deformation occurs

Perfectly inelastic collisions (web demos)
– objects stick together after the collision
Using conservation of momentum
• We can predict the outcome of collisions
using conservation of momentum
Let’s look at some collisions of carts on an
airtrack
– The objects only exert forces on each other.
– There are no external forces (like friction) so
momentum is conserved:
total momentum before the collision
=
total momentum after the collision
More Inelastic collisions

Now lets look at some inelastic collisions and see
how mass and speed influence the resulting
motions:
– Big fish/little fish
– Rear end accident
– Diesel engine and flatcar

Looking at these we are ready to set straight a
common movie mistake about momentum:
Bouncing
• A ball of mass 1 kg and travelling at v = 1 m/s hits a wall:
Case A: ball bounces
before:
after:
Case B: ball doesn’t bounce
before:
after:
• In which case is the change in momentum larger?
• In which case does the wall supply a greater impulse?
• Let’s first consider a question about bouncing
• Now let’s look at a bullet hitting a wooden block with and
without bouncing.
2-D collisions & explosions

In 2D collisions we must take the vector nature of
momentum into account:
Combined momentum

Let’s look at some examples of 2D collisions:
– 2 objects in an elastic collision
– 2 cars in an inelastic collision
– And finally, let’s play some pool/billiards
Chapter 7: Energy
Chap 7: Work and Energy

Energy comes in many different forms:
–
–
–
–

energy associated with motion (kinetic)
energy associated with position (potential)
Chemical energy
Heat energy
We will focus on mechanical energy:
– kinetic and potential energies

We will consider a number of topics in the Chapter:
–
–
–
–
–
–
work done on objects
power (rate at which work is done)
different types of energy
how work and energy are related
conservation of energy
energy and momentum
Work

Last chapter we considered: How long a force is applied
Impulse = Force * time
 Now: How long measured in distance rather than time
Work = force * distance
W
UNITS:

=
[Joule, J] =
F *d
[N] . [m]
1 Joule of work is done by a force of 1 Newton exerted on an
object over a distance of 1 meter.
– 1 kiloJoule
= 1 kJ = 1000J
– 1 megaJoule
= 1 MJ = 1,000,000 J
Examples

Push on a stationary object
How much work is done
on the object if it remains at rest?


Push on a car that moves a distance, d
F
d
Now pull on the car to slow it down
F
Car moving this way
Example
A rope applies a horizontal force of 200N to a crate over
a distance of 2 meters across the floor. A frictional force
of 150 N opposes this motion.
(a) What is the work done on the crate by the rope?
(b) What is the work done by the frictional force?
(c) What is the work done by the support force and
the gravitational force on the crate?
(d) What is the total work done on the crate?
Work done when lifting objects

In order for you to lift something at a constant speed you must
exert a force equal to the gravitational force on the object (its
weight)
– You do positive work
– Gravity does negative work

What if the lifting is done at an angle rather than straight up?
For instance, if you push a block up an incline?
Ans: if you lift the block to the same final height, the work
done by gravity is the same in both cases and so the work you
do is the same in both cases!

What about the force you exert? In which case is it less?
Let’s look at an animation of this …..

Power

How fast is work being done?
Power = work done / time

UNITS: [J/s] = [Watt, W]
– named after James Watt, developer of the steam
engine
If you do 1 J of work in 1 second, you have used 1
Watt of power.

Again we have:
1 kW = 1000W
1 MW = 1,000,000 W
Power calculation examples
1.
If little Nellie Newton lifts her 40-kg body a
distance of 0.25 meters in 2 seconds, then what is
the power delivered by little Nellie's biceps?
2.
Two physics students, Albert and Isaac, are in the
weightlifting room. Albert lifts the 100-pound
barbell over his head 10 times in one minute;
Isaac lifts the 100-pound barbell over his head 10
times in 10 seconds.
–
Which student does the most work?
–
Which student delivers the most power?
Other common units of energy

Heat energy: In chemistry and in PHY 102 we will use:
1 calorie = 4.19 J

Food products: use energy units of: Calories
1 Calorie = 1 kilocalorie = 4190 J

Electricity bill:
– units of energy on bill are: kWhr
– 1 kWhr = kilowatt * hour = 3,600,000 J
– [energy] = [power] * [time]

A 75 W light bulb uses 75 J of energy per second. If you use the
bulb for 4 hours how much energy (in kWhr) do you use?
Another common unit of power
 Cars measure power in horsepower
1 horsepower = 746 Watts or about 0.75 kiloWatt

Origin of the term “horsepower”:
– Ironically, the term horsepower (hp) was
invented by James Watt! He made an
estimate of how much work one horse could
do in one minute:
33,000 foot-pounds of work / minute.
– So, for example, a horse exerting 1 hp can
raise 330 pounds of coal 100 feet in a
minute.
Example:
My Volvo has a power rating of 175 hp. If I bought it in
Sweden how would they advertise the power rating?
Mechanical Energy

What enables something to do work?
ENERGY

we will focus on two types of energy:
(1) Kinetic energy (K.E.)
– energy due to the motion of an object
(2) Potential energy (P.E.)
– energy due to the relative position of an object
Kinetic energy
“energy of motion”
 if v = 0
K.E. = 0
 if you push on an object then its velocity
increases
K.E. increases as well
 The relationship between the object’s
energy and its speed is given by:

K.E. = 1/2 * mass * (speed)2
= 1/2 m v2
Examples
1.
A dragonfly has mass m = 10 g = 0.01 kg and flies at a
speed v = 10 m/s. What is it’s K.E.?
KE = ?
2.
A truck has mass m = 2000 kg and v = 2.0 m/s. how much
K.E. does the truck have?
KE = ?
3.
4.
Determine the kinetic energy of a 1000 kg roller coaster car
that is moving with a speed of 20.0 m/s.
If the roller coaster car in the above problem were moving
with twice the speed, then what would be its new kinetic
energy?
Work & Energy: how are they related?

If a force does work on an object
it changes the energy of the object

Consider a force on a block mass, m, that moves the block a
F
distance, d:
d
–
–
–
–
Let’s start with Newton’s 2nd Law : Fnet = m a
This tells us that the object accelerates
If its speed increases, then so does its kinetic energy
It can be show that:
Work-energy theorem
Wnet = D K.E.
Work done on an object = change in object’s K.E.
Why net work: Wnet?

Consider a case where friction is present
f

Fnet = F - f
F
Wnet = Fnet d = D K.E.
– only part of the work done by the force F goes
into changing the bock’s K.E.
– rest of the energy is transformed into heat
energy which results from the friction
Example
A 1000 kg car moving at 10 m/s (36 km/h) skids 5.0m with
locked wheels (wheels not turning) before it stops. How far will
the car skid before it stops if it is initially moving at 30 m/s (108
km/h)?

We find that: (K.E)case 2 = 9 * (K.E)case 1
– but this still doesn’t give us the distance!

Work done by the stopping force (brakes) = F * d
– this does have distance information
So we need to use the Work - energy theorem
Lets see what this all looks like in action
Potential energy (P.E)

Can take on different forms:
– elastic potential energy:


stretched/compressed spring
stretched rubber band
– chemical potential energy in fuels
– gravitational potential energy
P.E. due to elevated position of an object
= work done on an object against gravity when lifting it
= F * d = (mg) * d (lifting at constant v)
F
= mass * gravity * height
P.E. = m * g * h
So P.E. is proportional to mass and height.
mg

Where do we measure height, h, from?
– h is really a change in height
– must specify a level relative to which we measure h



So it is actually better to think of a change in P.E. associated
with a change in h
In Diagram A we have chosen PE = 0 and h =0 at the bottom.
Then all PE values given after that are relative to that reference
level.
What if we chose instead PE = 0 at the level of the first step in
Diagram C. How would the numbers change?
Example

How much potential energy does a 100 kg mountaineer
gain when they climb Mt Everest
(8.84 km) if the mountaineer starts at sea-level?

What if the person starts at base camp at 6.0 km?
h = 8840 m
ho=6000m
ho = 0
Energy continued: Summary

Work = Force x distance [N.m = J, Joule]
– work can be positive (if F and d are in same direction)
– work can be negative (if F and d are in opposite directions)
– Work is zero if F and d are perpendicular to each other

Power: rate at which work is done
= (Work done) / (time interval) [ J/s = W, Watt]
– Power usage in the home

Mechanical energy
– Kinetic energy: KE = 1/2 m v2
– Potential energy
gravitational PE = mass x gravity x height
Conservation of Energy
Energy can’t be created or destroyed!
BUT
 Energy can be transformed from one form to another
Consider some examples:
 Pendulum:

E = P.E.
E = P.E.
E = P.E. + K.E.
E = K.E.
E always the same at each position
but can be P.E., or K.E.
or a combination of both

Diver: see Fig 7.10
 Cart on an Incline
 Projectile
 Roller coaster
 Ski jump:


Spring potential energy
Bungee jumper
What if friction is present?

In the previous examples we conserved mechanical energy
because we ignored friction and air drag.
 What if friction is present?
– Still have conservation of energy but now we conserve
total energy instead of mechanical energy
Example: When a skier comes down an icy slope (no friction)
and then hits a flat section of unpacked snow she slows
down. She loses KE because energy is transferred to the
snow as heat energy due to friction between her skies and
the snow.
Demo of this ……

More complex systems:
Chemical potential energy
Coal is removed
we burn it to make
electricity
Radiant energy
Provides energy (nutrients) for plant growth
ground
Plant matter is buried
Forms coal
Machines





Make life easier (and simpler?) for us
again we need to take into account that energy cannot
be created or destroyed
So: energy in = energy out
If there is no friction in the system (ideal machine):
work in = work out
(F * d )in = (F * d )out
Machines can do two things:
1. Fout > Fin (multiply the size of the force)
2. change the direction of the force

Example: simple lever can increase the size of the
force and change it’s direction
Efficiency

Most machines are not ideal
 This means that (F d )in > (F d )out
– in other words not everything you put into the machine
is available to do work at the output end
– some of the work you do (i.e. energy you put in) does
work against friction
– we generally consider this to be a “loss” because the
output from the machine is diminished
– of course, the energy is not really lost - it is transformed
into a form we don’t find as useful!!
Work out = efficiency * (work in)
Example
Gasoline contains P.E. = 154 MJ/gallon.
Suppose you could build a car with a 100% efficient
engine.
If at a cruising speed air drag and road friction
combine to give you f = 500N, what is your fuel
consumption?
In other words, how far can you drive on 1 gallon of
gas?
Kinetic Energy & Momentum

1.
Both momentum & K.E. energy are associated
with motion of an object, but:
They are not the same type of quantity:
– Momentum is vector
– Energy is a scalar
– What is the relevance of this?
– Momentum vectors can add together to give
you total momentum = 0
– Energies can never add up to zero!
2.
They have different dependences on velocity
– Momentum is proportional to v
– Kinetic energy is proportional to v2
Damage

Damage to one object is related to the kinetic
energy of the other object striking it.
– More KE means the striking object can do more work
on other object & therefore can deform it more.

Example: While playing football you tackle
another player in a head-on collision. Your
momentums are equal & opposite before the tackle
so you come to a dead stop.
Question: Which hurts more?
– To be hit by a fast moving light player?, or
– A slow moving heavy player?

Hint: Which player has more K.E.?
Chapter 8: Rotational Motion
Some questions you’ll be able to answer
after today






Why is it more fun to be on the outside of the
merry-go-round?
How do trains go round curves?
Why is it easier to balance a hammer when the
head is up?
Why do SUV’s roll more easily than cars?
Why do the clothes in the washer all end up on the
outer wall of the washer during the spin cycle?
How do ice skaters manage to spin so fast?
Some of these questions stated more scientifically:

How do rotational and linear speeds relate?
 What actually causes things to rotate?
 About what point do objects rotate?
 When you put an object down how can you
predict whether it will stand or fall over?
 What is the difference between centripetal
force and centrifugal force?
 If we have linear momentum (mass *
velocity) can we have a similar quantity for
rotational motion?
Rotational Speed

linear speed = distance covered per unit time
 with circular motion can have constant speed
but direction is changing


linear speed depends on
how far from axis
point of interest is
A
Rotational speed =
number of rotations per unit time
B
Rotational & Linear speeds


Rotational speed usually measured in:
RPM = rotations (or revolutions) per minute
Example: old vinyl LP
RPM = 33 1/3
How are rotational and linear speed related?
 Linear speed is proportional to:
– 1. rotational speed, and
– 2. distance from axis
Turning corners

Car wheels
– turn independently
– outside wheel turns faster than inside one

Train wheels
– fixed axis turns wheels at same rate!
– but train wheels are tapered
Narrow part of the wheel
has smaller linear speed
(less distance in same
amount of time)
Wide part of the wheel has a larger
linear speed
Rotational Inertia



Newton’s 1st Law = Law of inertia
Recall: mass is the measure of linear inertia
– the greater the mass (inertia) of an object the greater the
object’s resistance to change in motion (linear acceleration)
So: large inertia (mass)
small acceleration
now rotational inertia (I) is related to mass distribution
– Ex:
most of the mass
of the bat is here
If you hold bat here
large I
If you hold bat here
smaller I
Rotational Inertia by shape

Calculating rotational inertias is tricky, but what
we can do is notice that I depends on:
(1) shape of the object
(2) the axis of rotation you choose

Ex: Look at the hoop

If most of the mass is located far from axis
object has a large I
 Now just as for linear motion:
– the greater the rotational inertia of an object the
greater the object’s resistance to change in rotational
motion (rotational acceleration)
So: large I
small rotational acceleration
Rotation



Why do we get a rotation?
Consider linear motion:
– unbalanced force causes a change in linear motion
What is the rotational equivalent?
– unbalanced torque causes a change in rotational motion
Torque = lever arm x force
Vector: has magnitude and direction
(we will describe direction in terms
of the rotation it causes: clockwise or
counterclockwise)
Distance from the axis
of rotation to where
the force is applied
Example: seesaw
– consider the torques exerted by the boy and girl on
the seesaw
– net torque = the sum of the individual torques
If net torque = 0 then there are no
unbalanced torques and so no rotation!
 Let’s
look at a Web demo
 Now try Practice Page 31 (mobile)
Center of mass


spin something: it seems to rotate about a specific point.
Let’s go back to projectile motion:
– throw a ball and it follows a parabolic path
– Now throw a baseball, what path does it follow?
How does the bat move?
– What if you spin a wrench across a frictionless table? How
does it move?

All these objects rotate about the “center” of the
object
– not a geometric center but rather the:
“center of mass” = average position of all the mass
that makes up the object

Object’s motion can then be separated into:
– linear motion
– rotational motion

to determine the linear motion of object pretend all
the mass of the object is located at the center of
mass
Center of mass vs Center of gravity

For our purposes:
Center of mass (CoM) = Center of gravity (CoG)

if gravity (g) is constant everywhere in the object
then CoM & CoG are located at the same point

CoM & CoG are not in the same location if the object
is very large (then g varies across the object)
How do we find the CoG?
Let’s consider a few different methods:
1. Symmetric objects: find geometric center
– if object is symmetric & has uniform density, then:
geometric center = center of gravity
2. Find the balance point of the object:
3. Suspend the object:
CoG located where
the 2 lines cross
Examples

Where are the CoG’s located for:

Ex 1: Donut

Ex 2: L-shape

Ex 3: Web Demo: Explorelearning
Stability

When will an object stand & when will it topple?

Most of us can tell this intuitively but what rule
would you give someone?
 Object is supported by its base

See where the CoG is located relative to base:
– if the CoG is located above the base = stable
– if the CoG is not over the base = unstable
CoG
Circular motion

Spin a ball on a string
– what happens if string snaps?
– what causes the ball to move
in a circle?

The string provides a centripetal force

What is a centripetal force?
– Any force that causes circular motion
 tension force from string
 gravitational force (moon orbits earth in a circular path)
 what force keeps a car on a circular track?
Centrifugal force?

So what is this centrifugal force that so many people talk
about?

Centrifugal = center fleeing, away from center
– this is an “apparent” force

When a car turns corner what happens?
– The frictional force between car & road causes a
centripetal force on car (so the car turns)
– no seatbelt & slippery seats in your car: you keep going
in a straight line
– it appears as if there is an outward (centrifugal) force
acting on you
– the centrifugal force is actually a lack of a centripetal
force on you!
Web Link: Right Hand Turn
Example: Amusement park ride
Spins fast


you feel like you are pushed outward
let’s look at forces acting on you:
Force of wall
pushing on you
This is centripetal force
that makes you turn
Friction, f
Weight, W = mg
From your perspective you feel an outward: centrifugal force
Angular momentum

linear momentum =

angular momentum = rotational inertia * rotational velocity
=
mass * velocity
I
*
w
An object’s linear momentum changes only if a force acts
on it
 an object will change its angular momentum only if an
external torque acts on it

Conservation of momentum

conservation of linear momentum
– if no external force acts on system then linear
momentum is conserved

conservation of angular momentum
– if no external torque acts on system then angular
momentum is conserved

linear case:
 angular case:
(mv)before
(I w)before
Web Link: Merry-go-Round
=
=
(mv)after
(I w)after

Radians and pi (p)
Sometimes angles are measured in degrees
– Ex: 90o, 45o, etc

Can also measure angles in radians [rad]
 How do we define radians?
–
–
–
–
–
–
One complete rotation = 360o
circumference of a circle: C = 2 p R
if set R = 1, then C = 2 p
so distance covered in 1 rotation = 2 p
say 2 p radians = 360o
or 1 radian = 360o / 2 p
R