PHY 101 PHY 101 Lecture NotesLecture Notes

Instructor: Laura Fellman

Chapter 2Chapter 2

A brief look at the historical A brief look at the historical development of physics and development of physics and Newton’s 1Newton’s 1stst Law of Motion Law of Motion

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!

Aristotle (384-322 BC)Aristotle (384-322 BC)

Nicolaus Copernicus (1473-1543)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)Galileo Galilei (1561 – 1642)professor of Mathematics at University in Italy Galileo used observations and experiments to disprove Aristotle’s ideashe was interested in HOW things moved, not why they moved.we call this kinematicsImportant experiment: Galileo dropped heavy

and light objects together and found they hit the ground at the same time.– See the experiment in action

Air resistanceAir 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 GalileoBack 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 telescopeGalileo & 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 troubleGalileo 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)Newton(1642-1727)

Changed the focus from “how” to “why”Made brilliant contributions to physics!Pondered why apple fell to

Earth amongst other thingsHe summarized his findings

in 3 laws = Newton’s LawsAll involve the idea of a force

(or lack of a force)

Isaac Newton: Yes, the apple Isaac Newton: Yes, the apple really fell!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 InertiaNewton’s First Law/ Law of InertiaAn 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.

ForceForce 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 forceNet force

Need to combine the forces & find net force

Fnet ?

Fnet ?

Fnet ?

3N

2N

2N

2N

3N4N

2N

Review of Law of InertiaReview of Law of Inertia

See this online summary

EquilibriumEquilibrium Condition for equilibrium: Fnet = 0

– so all forces balance each other

Static equilibrium: speed = 0 (no motion), and

Fnet = 0

Support forcesQ. What stops a book from falling through the

table it lies on?Ans: A support or “Normal” force

What’s normal about it?

Examples: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?

TensionTension 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

1

2

3T1 , T2 and T3 are all equal in size, but in different directions.

Examples:Examples:Window washers: Joe and Jane (equal weights)

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 scaleT1 = ? T2 = ?

spring scaleT1 = ? T2 = ?

Let’s try practice pages 3 and 4 now in your Practicing Physics book

Then we’ll try this question……

Dynamic equilibriumDynamic equilibriumConditions 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 MotionChap 3: Linear Motion

Let’s find ways to describe Let’s find ways to describe howhow things movethings move

Description of MotionDescription 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]

0 1 2 3 4

time (t) in [seconds, s]

See motion being graphed in passing lane demo

4

3

2

1

Distance and timeDistance 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?Speed: how fast?

distance speed =

time

Units: km/hour or mph or m/s

Two ways to look at speed:

(1) average speed

(2) instantaneous speed

SI Unit for speedSI Unit for speed

Average speedAverage 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:

total distance coveredaverage speed =

time interval

Instantaneous speedInstantaneous speedSpeed at any one instantExample: 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 timeGraphing speed vs time Just like we graphed position vs time, we can graph velocity as it

changes with time.

position (x)

in [m]

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.

4321

4321

velocity (v)

in [m/s]

Examples involving distance and speedExamples involving distance and speed

Let’s try some conceptual questions:

– Motorist

– Bikes and Bees

More on average speedMore 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?

VelocityVelocityNow we consider speed and directionExample:

– 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 VelocitySpeed 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

AccelerationAcceleration Acceleration = rate of change of velocity

= change of velocity

time interval

acceleration: speeding up or slowing down

Q. Can we feel velocity?

Q. Can we feel acceleration?

Q. What controls in a car make it accelerate?

ExamplesExamplesEx 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.stime 10 s

Ex 2: A cyclist’s speed increases from 4 m/s to 10 m/s in 3 seconds.

Q. What is the cyclist’s acceleration?

Graphs showing accelerationGraphs 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. position vs time

2. velocity vs time

3. acceleration vs time

Acceleration on inclined planesAcceleration 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 C

(Hint: see Fig 3.6 in textbook)

Free fallFree 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?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 relationshipsSummary: 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 speedv = 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) = ½ g * t 2 (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.

ExamplesExamples

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 2Chapter 4: Newton’s 2ndnd Law Law

WhyWhy things move 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:Notation:

Weight(gravitational force)

W

NNormal force(contact force)

FPulling or pushing force

Example:Example: If the block has a mass of 10 kg and if pulled by 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 a force of 50N, find the values of the forces shown in the above diagram and calculate the horizontal acceleration. above diagram and calculate the horizontal acceleration.

Rank the accelerations, smallest to largestRank the accelerations, smallest to largest

A B C

D

Mass, Weight & VolumeMass, 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)

ExamplesExamples 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 2Newton’s 2ndnd Law in many object problems 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?

2kg 2kg 2kgTT11 = 30N = 30N

TT22 = ? = ?TT33 = ? = ?

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

FrictionNeed: 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? • in opposite direction to motion

• What causes the friction?– Irregularities (roughness) in surfaces

Direction of motion

Frictional force

2 kinds of friction• Static friction: before there is any motion

• Sliding friction: when block is motion

• Static friction > sliding friction

Friction = 50N Applied force = 50N

v = constant

Friction = 70N Applied force = 70N

v = 0

Interesting facts about frictionInteresting 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 & frictionForce at angles & friction

frictionfriction

support forcesupport force

tensiontension

weightweight

horizontal part of tensionhorizontal part of tension

verticalverticalpart of part of tensiontension

Free-fall revisitedFree-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)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 fallNon-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 velocityTerminal 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 examplesLet’s consider some examples

First let’s look at the force of air resistance Let’s revisit the elephant and the feather

fallingNow 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 Chapter 5: Newton’s 3rd LawLawandand

VectorsVectors

Newton’s 3rd Law Newton’s 3rd Law

Whenever object A exerts a force on object Bobject B, object Bobject 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 ProblemsExamples and ProblemsFirst we’ll consider the question of an apple

on a table.Now look at these examples of force pairsAnd 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 systemDefining 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 2Combining Newton’s 2ndnd and 3 and 3rdrd laws laws

Example 1:

Find: (a) acceleration of the blocks

(b) force on block B by block A

(c) force on block A by block B

5 kg10 kg

AABB

150 N150 N

Ice, so no friction to worry aboutIce, so no friction to worry about

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 Vectors2D motion and Vectors

2-D Motion2-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 MotionRelative Motion

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

tailwind

headwind

tailwind

headwind

VectorsVectors 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)Vector example (1-D) 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

– Now consider a headwind: vw = 20 km/h to south

vR = vA + vw = 100 km/h + (-20 km/h) = 80 km/h

wind

planeresultant

wind

planeresultant

Crosswind (2-D)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 MotionChapter 10: Projectile Motion

a.k.a. How to hit your neighbors with a cannon ball!a.k.a. How to hit your neighbors with a cannon ball!

?

Projectile motionProjectile 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 Target practice: Horizontal LaunchLaunch

• 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 Now add in the velocity componentsvelocity components

Firing at an angleFiring 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 projectileVelocity 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?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 rangerange 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:

RangeRange = 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 1Example 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?

20 m

1.5 m

Example 2Example 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?

500 m?

40 m/s

Example 3Example 3A 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 4Example 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: MomentumChapter 6: Momentum

Chapter 6 : MomentumChapter 6 : Momentum Momentum is inertia (m) in motion (v)

momentum = mass * velocity

p = m * v

UNITS: kg m /s (no special name)

Values of momentumValues of momentum

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)

m = 7 kgm = 7 kg

p = 14 kg m /sp = 14 kg m /sv = 2 m/sv = 2 m/s

m = 0.070 kgm = 0.070 kg

p = 14 kg m /sp = 14 kg m /s

v = 200 m/sv = 200 m/s

Force & MomentumForce & 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 momentumHow 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 momentumChanging 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:

When will final velocity be greater:short push or long push?

F

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 examplesSome more examples1. 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 momentumConservation 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 Example

Rifle fires a bullet or a cannon fires a cannonball– Web demo: cannonball fired

1. Draw situation beforebefore

the action (firing of rifle):

2. Draw the situation afterafter 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:(momentum of system)before = (momentum of system)after

m

M

More Examples 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: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?

CollisionsCollisions 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 Using conservation of momentummomentum

• 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 collisionsMore 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:

BouncingBouncing• A ball of mass 1 kg and travelling at v = 1 m/s hits a wall:

Case A: ball bouncesbefore: after:

Case B: ball doesn’t bouncebefore: 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 & explosions2-D collisions & explosions In 2D collisions we must take the vector nature of

momentum into account:

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

Combined momentum

Chapter 7: EnergyChapter 7: Energy

Chap 7: Work and EnergyChap 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 energymechanical 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

WorkWork

Last chapter we considered: How long a force is appliedImpulse = Force * time

Now: How long measured in distance rather than timeWork = force * distance W = F * d

UNITS: [Joule, J] = [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

ExamplesExamples

Push on a stationary objectHow much work is done

on the object if it remains at rest?

Push on a car that moves a distance, d

Now pull on the car to slow it down

Fd

FCar moving this way

ExampleExample

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 objectsWork 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:Ans: if you lift the block to the same final height, the work done by gravity is the samesame in both cases and so the work you work you do is the same in both casesdo 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 …..

PowerPower 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 examplesPower calculation examples1. 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 energyOther common units of energy Heat energy: In chemistry and in PHY 102 we will use:

1 calorie = 4.19 J1 calorie = 4.19 J

Food products: use energy units of: Calories1 Calorie = 1 kilocalorie = 4190 J1 Calorie = 1 kilocalorie = 4190 J

Electricity bill:– units of energy on bill are: kWhr kWhr – 1 kWhr = kilowatt * hour = 3,600,000 J1 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 powerAnother common unit of power

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.

Cars measure power in horsepower 1 horsepower = 746 Watts1 horsepower = 746 Watts or about 0.75 kiloWatt

Example:My Volvo has a power rating of 175 hp. If I bought it in Sweden how would they advertise the power rating?

Mechanical EnergyMechanical Energy What enables something to do work?

ENERGY

we will focus on two types of energy:

(1) Kinetic energy (K.E.)Kinetic energy (K.E.)– energy due to the motion of an object

(2) Potential energy (P.E.)Potential energy (P.E.)– energy due to the relative position of an object

Kinetic energyKinetic energy“energy of motion”if v = 0 K.E. = 0if you push on an object then its velocity

increases K.E. increases as wellThe relationship between the object’s

energy and its speed is given by:

K.E.K.E. = 1/2 * mass * (speed)= 1/2 * mass * (speed)22

= 1/2 m v= 1/2 m v22

ExamplesExamples1. 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.?

2. A truck has mass m = 2000 kg and v = 2.0 m/s. how much K.E. does the truck have?

3. Determine the kinetic energy of a 1000 kg roller coaster car that is moving with a speed of 20.0 m/s.

4. If the roller coaster car in the above problem were moving with twice the speed, then what would be its new kinetic energy?

KE = ?

KE = ?

Work & Energy: how are they related?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 distance, 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:

Wnet = K.E.

Work done on an object = change in object’s K.E.Work done on an object = change in object’s K.E.

Fd

Work-energy theoremWork-energy theorem

Why Why netnet work: W work: Wnetnet??

Consider a case where friction is present

Fnet = F - f Wnet = Fnet 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

Ff

ExampleExample

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)Potential energy (P.E) Can take on different forms:

– elastic elastic potential energy: stretched/compressed spring stretched rubber band

– chemical chemical potential energy in fuels– gravitational 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)

= mass * gravity * height

P.E. = m * g * h

So P.E. is proportional to mass and height.

mg

FF

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?

ExampleExample 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?

ho = 0

h = 8840 m

ho=6000m

Energy continued: SummaryEnergy continued: Summary

Work = Force x distanceWork = 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

PowerPower:: rate at which work is done

= (Work done) / (time interval) [ J/s = W, Watt]

– Power usage in the homePower usage in the home

Mechanical energyMechanical energy– Kinetic energy: KE = 1/2 m v2

– Potential energy

gravitational PE = mass x gravity x height

Conservation of EnergyConservation of Energy

Energy can’t be created or destroyed!

BUT Energy can be transformedtransformed from one form to another

Consider some examples:Consider some examples: Pendulum:

E = P.E.

E = K.E.

E = P.E.

E = P.E. + K.E. E always the same at each positionbut 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?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

Radiant energy

Provides energy (nutrients) for plant growth

groundPlant matter is buried

Forms coal

Coal is removedwe burn it to make electricity

MachinesMachines 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 outenergy 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

EfficiencyEfficiency 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 = Work out = efficiencyefficiency * (work in) * (work in)

ExampleExample

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 & MomentumKinetic Energy & Momentum Both momentum & K.E. energy are associated

with motion of an object, but:1. 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

DamageDamage 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?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 MotionChapter 8: Rotational Motion

Some questions you’ll be able to answer Some questions you’ll be able to answer after todayafter 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: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 SpeedRotational Speedlinear speed = distance covered per unit timewith circular motion can have constant speed

but direction is changing

linear speed depends on how far from axis point of interest is

Rotational speed = number of rotations per unit time

A B

Rotational & Linear speedsRotational & Linear speedsRotational 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 cornersTurning cornersCar 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 sameamount of time)

Wide part of the wheel has a larger linear speed

Rotational InertiaRotational 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:

If you hold bat here smaller I

If you hold bat here large I

most of the massof the bat is here

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

Rotational Inertia by shapeRotational Inertia by shape

RotationRotation 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 axisof rotation to wherethe force is applied

Example: seesawExample: 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 demoNow try Practice Page 31 (mobile)

Center of massCenter 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 gravityCenter 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?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

ExamplesExamplesWhere are the CoG’s located for:

Ex 1: Donut

Ex 2: L-shape

Ex 3: Web Demo: Explorelearning

StabilityStability 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 motionCircular motionSpin 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?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 rideExample: Amusement park ride

you feel like you are pushed outward let’s look at forces acting on you:

Spins fast

Friction, f

Weight, W = mg

Force of wall pushing on you

This is centripetal force that makes you turn

From your perspective you feel an outward: centrifugal force

Angular momentumAngular momentumlinear momentum = mass * velocity

angular momentum = rotational inertia * rotational velocity

= I *

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 momentumConservation 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: (mv)before = (mv)after angular case: (I )before = (I )after

Web Link: Merry-go-Round

Radians and pi Radians and pi Sometimes angles are measured in degrees

– Ex: 90o, 45o, etcCan also measure angles in radians [rad]How do we define radians?

– One complete rotation = 360o

– circumference of a circle: C = 2 R– if set R = 1, then C = 2 – so distance covered in 1 rotation = 2 – say 2 radians = 360o

– or 1 radian = 360o / 2

R