General Physics I

Chapter 1 Measurement

Lecture Notes

Observations• The sciences are ultimately based on observations of the natural (&

unnatural) world• There are 2 types of observations:• Qualitative

– Subjective, touchy-feely Example: the outside temperature is hot today

• Quantitative– Objective, based on a number and a reference scale– Quantitative observations are referred to as measurementsExample: the outside temperature is 80oF today

Notes:1. Quantitative observations are only as reliable as the measurement device and

the individual(s) performing the measurement

2. The accuracy associated with a measurement (or set of measurements) is the often specified as the % Error:

3. The precision associated with a set of measurements is the often specified as the % Range:

%accepted value - measured value

% Error = 100accepted value

%highest value - lowest valueRange

% Range = 100 %= 100average value average value

Systems of Measurement• There are several units systems for measurement of physical

quantities• The most common unit systems are the metric and the USCS systems• For consistency, the l’Systeme Internationale (or SI) was adopted

– The SI system is a special set of metric units• International System (SI) base units:

Mass Kilogram kgLength meter mTime second sTemperature Kelvin KCurrent Ampere ALuminous Intensity candela cdAmount of substance mole mol

• All of the other SI units are derived from these base units Examples of derived units:

1 Newton = 1 N = 1 kg.m/s2

1 Joule = 1 J = 1 kg.m2/s2

1 Coulomb = 1 C = 1 A.s

Common Metric Prefixes

Using Metric prefixes:1 mm = 1x10-3 m 35 mm = 35 x 10-3 m or 3.5 x 10-2 m1 kg = 1x103 g 12 kg = 12 x 103 g or 1.2 x 104 g

Prefix Symbol Meaning Power of 10

Giga G 1,000,000,000 109

Mega M 1,000,000 106

kilo k 1,000 103

centi c 0.01 10-2

milli m 0.001 10-3

micro 0.000,001 10-6

nano n 0.000,000,001 10-9

Unit ConversionIn physics, converting units from one unit system to another (especially

within the Metric system) can appear daunting at first glance. However, with a little guidance, and a lot of practice, you can develop the necessary skill set to master this process

Example: How is 25.2 miles/hour expressed in m/s? 1. Eliminate: {assign mi units to the denominator and hr units to the numerator of

the conversion factor}

2. Replace: {assign m units to the numerator and s units to the denominator of the conversion factor}

3. Relate: {assign the corresponding value to its unit, 1 mi = 1609 m & 1 hr = 3600 s}

.25.2 mi ?? m ?? hr m

25 2 = ?? 1 hr ?? mi ?? s s

mihr

.25.2 mi ?? ?? hr m

25 2 = ?? 1 hr ?? mi ?? s

mihr

.25.2 mi 1609 m 1 hr m

25 2 = 11.3 1 hr 1 mi 3600 s s

mihr

Length, Time & MassLength is the 1-D measure of distance• Quantities such as area and volume, and their associated units, are ultimately

derived from measures of length

Definition of SI Unit: The meter is the length of the path traveled by light (in vacuum) during a time interval of 1/299,792,458 s (or roughly 3.33564 ns)Examples of units derived from length (in this case radius, r):

1. Area of a circle: {units are m2}

2. Volume of a sphere: {units are m3}

Time is the physical quantity that measures either:__ when an event took place__ the duration of the event

Definition of the SI Unit: The second is the time taken by 9,192,631,770 oscillations of the light emitted by a cesium-133 atom

Mass is the measure of inertia for a body (or loosely speaking the amount of matter present)Definition of the SI Unit: The kilogram is the amount of mass in a platinum-iridium cylinder of 3.9 cm height and diameter.

34sphere 3V = r

2circleA = r

Trigonometry(remember: SOHCAHTOA)

• The relationships between sides and angles of right triangles• Consider the following right triangle:

The 3 primary relations between the sides and angles are:

1. The Sine:

2. The Cosine:

3. The Tangent:

AC

oppositesin(angle) =

hA

sypot

inenuse

= C

adjacentcos(angle) =

hB

cypot

osenuse

= C

oppositetan(angle) =

adjaA

tancent

= B

Vectors & ScalarsMost physical quantities can categorized as one of 2 types (tensors

notwithstanding):1. Scalars:

• described by a single number & a unit (s).Example: the length of the driveway is 3.5 m

2. Vectors:• described by a value (magnitude) & direction.Example: the wind is blowing 20 m/s due north– Vectors are represented by an arrow:

a. the length of the arrow is proportional to the magnitude of the vector.

b. The direction of the arrow represents the direction of the vector

• Only vectors of the same kind can be added together • 2 or more vectors can be added together to obtain a “resultant” vector• The “resultant” vector represents the combined effects of multiple vectors

acting on the same object/system– Direction as well as magnitude must be taken into account when adding

vectors– When vectors are co-linear they can be added like scalars

Properties of Vectors

A B R+ =

A B R+ =

• Any single vector can be treated as a “resultant” vector and represented as 2 or more “component vectors

• To add vectors of this type requires sophisticated mathematics or use of graphical techniques

Ax

AyA

+=Ax

Ay=

Vector AdditionA. Graphic Method• To add 2 vectors, place them tail-to-head, without changing their direction; the

sum (resultant) is the vector obtained by connecting the tail of the first vector with the head of the second vector

• R=A+B means “the vector R is the sum of vectors A and B”

• R A+B : the magnitude of the vector R is NOT equal to the sum of the magnitudes of vectors A and B

R2 = A2 + B2+ 2AB cos

• Note:– For co-linear vectors pointing in the same direction, R = A + B– For co-linear vectors pointing in opposite directions, R = A - B

R

A

B

Vector Addition (cont.)B. Component Method• Express each vector as the sum of 2 perpendicular vectors. The

direction of each component vector should be the same for both vectors. It is common to use the horizontal and vertical directions (These vectors are the horizontal and vertical components of the vector)

Example:

vector A Ax (horizontal) and Ay (vertical) or

vector B Bx (horizontal) and By (vertical) or

Note: The unit vectors x and y indicate the directions of the vector components

• The common component vectors for A & B can now be added together like scalars to obtain the component vectors for the resultant vector:

Rx = Ax + Bx and Ry = Ay + By thus:

ˆ ˆx y x yA = A + A = A x + A y

ˆ ˆx y x yB = B + B = B x + B y

ˆ ˆx y x yR= R +R = R x + R y

Vector Addition (cont.)• To calculate the magnitude of the resultant from the component vectors

by using the Pythagorean Theorem:

• The angle of the resultant vector (from the x axis) obtained from the ratio of component vectors:

• To calculate the components from the magnitude R and the angle it makes with the horizontal direction:

xx

Rcos = R = Rcos

R

yy

Rsin = R = Rsin

R

2 2 2 2 2x y x yR = R + R R = R + R

Rx

RRy

x

y

y y-1

x x

R Rtan = = tan

R R

J. Willard Gibbs (1839-1903)

• American mathematician & physicist• Considered one of the greatest scientists of the

19th century• Major contributions in the fields of:

– Thermodynamics & Statistical mechanics• Formulated a concept of thermodynamic equilibrium of

a system in terms of energy and entropy

– Chemistry• Chemical equilibrium, and equilibria between phases

– Mathematics• Developed the foundation of vector mathematics

"A mathematician may say anything he

pleases, but a physicist must be at least partially sane."

14

General Physics I

Chapter 2: Motion along a Straight Line

Lecture Notes

15

Galileo Galilei (1564-1642)

• Credited with establishing the “scientific method”

• Based his scientific hypotheses on observation and experimentation

• First to use telescope for astronomical observation– Observed the following:

• The craters and features on the Moon• The moons of Jupiter & the rings of Saturn• Sun spots

– Based on his observations supported Copernican Theory

• Conclusively refuted Aristotelian ideology (and contradicted Church doctrine)

– Placed under “house arrest” as punishment

• Studied accelerated motion and established the first equations of kinematics

• Proposed the Law of Inertia, which later became known as Newton’s 1st law of Motion

All truths are easy to understand once they are discovered; the point is to

discover them.

16

Displacement• Motion: An object is in motion if it changes its position

relative to a reference point (Origin). Motion is relative.• Position (R) a vector that connects an object’s location

to a reference point (origin). As an object moves, the its position changes.

• Displacement (R) is the difference between the final position and the initial position of a moving object; its magnitude is measured in meters (m)

Origin (0,0)

R1 (position vector at t1)

R2 (position vector at t2)

R (Displacement)

1 2

17

Speed & VelocityVelocity is a vector with the same direction as the displacement vector

(R). Its magnitude is called speed and it is measured in m/s.

• Average Velocity:

• Average Speed:

• Instantaneous Velocity:

• Instantaneous Speed:

avg

displacement rv = =

time t

t 0

rv=lim

t

Origin (0,0)

R1

R2

R (Displacement)

1 2v (velocity)

v = v ="the magnitude of the velocity"

avg avg

distance traveledv = v =

t

18

Acceleration• When the velocity vector is changing in time, the acceleration vector

describes the time rate of change of the velocity vector.• Acceleration (a) is a vector with the same direction as the change in

velocity (v) vector. It is measured in m/s2.

• Average Acceleration:

• Instantaneous Acceleration:

Example 1: A car accelerates from 0 to 30 m/s in 6 s. Find the average acceleration.

Since v = 30 m/s,

t 0lim

va

t

_ _avg

change in velocity va

time t

2

305

6 s

ms

avg

v ma

t s

19

Equations of KinematicsWhen to = 0 (and a is constant), the most relevant equations of motion can

be written in scalar form as:

1. v1 - vo = at {Velocity Equation}

2. r1 - ro = vot + ½ at2 {Displacement Equation}

3. r1 - ro = ½(vo + v1)t {Average Velocity}

4. v12 - vo

2 = 2a(r - ro) {Galileo’s Equation}

These equations will allow us to completely describe the motion of a moving object and represent our “toolbox” for studying kinematics!

Example: A ball rolls along a flat surface with an acceleration of +1.0 m/s2. The initial position and velocity of the ball are xo=1m & vo= -2 m/s, respectively.

1. How fast is the ball traveling after 2 s have elapsed?

2. What is the displacement & final position of the ball after 2 s?

3. How fast is the ball traveling when it travels a displacement of 5 m?

4. How long does it take the ball to travel a displacement of 5 m?

20

Free Falling Bodies• The 1-D motion of objects subject to gravity• The only motion considered is vertical (use y for position)

• There is no air resistance• Acceleration of falling object is constant:

ay = -g = -9.8 m/s2

Example 1: Consider a rock dropped from rest.1. What is the velocity after the rock has fallen 3 s?2. How far does the rock travel during this 3 s?3. How fast is the rock moving after it has fallen 10 m?

Example 2: Consider a rock tossed into the air with an initial upward velocity of 5 m/s.

1. How long would it take for the rock to return to the thrower (initial height)?

2. What is the highest position the rock reaches during its ascent?

ay = 9.8 m/s2

21

Graphical Analysis of Velocity & Acceleration• position vs. time graph

– The slope is the velocity

• velocity vs. time graph– The slope is the acceleration

– The area under the curve is the displacement

22

General Physics I

Chapter 3: Motion in 2 Dimensions

Lecture Notes

23

2-Dimensional Motion

• A motion that is not along a straight line is a two-dimensional motion.

• The position, displacement, velocity and acceleration vectors are necessarily not on the same directions; the rules for vector addition and subtraction apply

• The instantaneous velocity vector (v) is always tangent to the trajectory but the average velocity (vavg) is always in the direction of R

• The instantaneous acceleration vector (a) is always tangent to the slope of the velocity at each instant in time but the average acceleration (aavg) is always in the direction of v

RB(t1)

RA(t0) v(t)

24

Example of 2-D Motion

25

Displacement, Velocity & Accelerations• When considering motion problems in 2-D, the definitions for the motion

vectors described in Chapter 2 still apply.• However, it is useful to break problems into two 1-D problems, usually

– Horizontal (x)

– Vertical (y)

• Displacement:

• Velocity:

• Acceleration:

• Use basic trigonometry relations to obtain vertical & horizontal components for all vectors

ˆ ˆyxdrdrdr

v= = i+ jdt dt dt

ˆ ˆyxdvdvdv

a= = i+ jdt dt dt

ˆ ˆo o or=r- r = x-x i+ y-y j

26

When to = 0 and a is constant:

Remember: when to≠0, replace t with t in the above equations

Equations of Kinematics in 2-D

Horizontal:

1. vx - vox = axt

2. x - xo = ½ (vox + vx)t

3. x - xo = voxt + ½ axt2

4. vx2 - vox

2 = 2axx

Vertical:

1. vy - voy = ayt

2. y - yo = ½ (voy + vy)t

3. y - yo = voyt + ½ ayt2

4. vy2 - voy

2 = 2ayy

Projectile Motion is the classic 2-D motion problem– Vertical motion is treated the same as free-fall

ay = - g = - 9.8 m/s2 {downward}– Horizontal motion is independent of vertical motion but connected by

time but no acceleration vector in horizontal direction

ax = 0 m/s2

– As with Free Fall Motion, air resistance is neglected

27

Projectile Motion Notes

t

x

t

y

t

vx

t

vy

t

x

t

y

t

vx

t

vy

28

Projectile Motion Notes (cont.)

29

General Physics I

Chapter 4: Forces & Newton’s Laws of Motion

Lecture Notes

30

Newton’s 1st Law

• Also known as the “Law of Inertia” • Key Points:

– When an object is moving in uniform linear motion it has no net force acting on it

– When there is no net force acting on an object, it will stay at rest or maintain its constant speed in a straight line

• This property of matter to maintain uniform motion is called inertia

• Stated a simpler way: Nature is lazy! {i.e. matter resists changes in motion}

F=0 a = 0 {v = constant}

Every object continues in its state of rest, or of motion unless compelled to change that state by forces impressed upon it.

31

Newton’s 2nd Law

• When a net force is exerted on an object its velocity will change:

• The time rate of change of motion (acceleration) is related to:– Proportional to the size of the net force– Inversely proportional to the mass of the object (i.e. its inertia)

• The relationship between them is

or

• The direction of will always correspond to the direction of

ˆ ˆx y

F 1a= = F x + F y

m m

ˆ ˆ

x yF = ma= m a x + a y

...1 2 3F=F F F dv

dt

a

F

32

Newton’s 3rd Law

When an object exerts a force on a second object, the second object exerts an equal but oppositely directed force on the first

object

where:

Consequences:• Forces always occur in action-reaction pairs (never by themselves)• Each force in an action-reaction pair acts on a different object

Important: • Newton’s 3rd law identifies the forces produced by interactions

between bodies• Newton’s 2nd law defines the accelerations that each object

undergoes

1 on 2FBody

1Body

22 on 1F

2 on 1 1 on 2F F

33

Free-Body Diagrams

• Simplified drawing of a body with only the forces acting on it specified

• The forces are drawn as vectors• Free-Body diagrams facilitate the application of Newton’s

2nd Law

Examples:

Joey

W

Joey in “Free Fall”

Joey

W

floorF

Joey standing on a floor

Joey

W

floorF

Joey getting slapped while standing on a floor

slapF

34

Types of Forces

Non-Contact:• Gravitational• Electric• Magnetic

Contact:• Normal• Frictional• Tension

In our world, forces can be categorized as one of 2 types:

• Non-Contact: force is exerted over a distance of space with out direct contact (a.k.a. “action-at-a-distance” forces)

• Contact: forces is exerted due to direct contact

(Note: at the microscopic level, ALL forces are non-contact)

• In either case, Newton’s 3rd law still applies to the forces present

Examples of each type of force:

35

Mass vs. Weight• The “weight” of an object is the gravitational force

exerted on it by the gravitational attraction between the object and its environment:

• On the surface of the Earth, the gravitational force is referred to as weight:

• Mass is a measure of an object’s inertia (measured in kg)

– Independent of object location• Weight is the effect of gravity on an object’s mass

(measured in N)– Determined by the local gravitational acceleration

surrounding the object

ˆG G G G G G GF =ma =ma y F =m a F =ma

ˆG GF = W = (-mg)y F = W = mg

Notes:• Mass is a measure of an object’s inertia (measured in kg)

– Independent of object location

• Weight is the effect of gravity on an object’s mass (measured in N)

– Determined by the local gravitational acceleration surrounding the object

36

Normal Force

• The “support” force between 2 surfaces in contact• Direction is always perpendicular (or normal) to the plane of the

area of contactExample: the force of floor that supports your weight

Consider standing on a scale on the floor of an elevator. The reading of the scale is equal to the normal force it exerts on you:Task: Construct free body diagrams for the scale:

1. At rest2. Constant velocity3. Accelerating upward4. Accelerating downward

37

Examples of Normal Force

38

Surface Frictional Forces• When an object moves or tends to move along a surface, there

is an interaction between the microscopic contact points on the 2 surfaces. This interaction results in a frictional force, that is– parallel to the surface– opposite to the direction of the motion

• There are 2 types of surface friction:– Static (sticking)– Kinetic (sliding)

39

Static Friction• Static (or sticking) friction ( ) is the frictional force

exerted when the object tends to move, but the external force is not yet strong enough to actually move the object.

• Increasing the applied force, the static frictional force increases as well (so the net force is zero) . The force just before breakaway is the maximum static frictional force.

• The direction of the static friction force is always in opposition to the external forces(s) acting on the body

• The magnitude of the maximum static friction force is:

Where:– is the coefficient (maximum) of static friction– FN is the normal force

max maxs s Nf =μ F

sf

maxsμ

40

Kinetic Friction

• Kinetic frictional force ( ) is the frictional force exerted by the surface on an object that is moving along the surface

• Kinetic frictional force:– always opposes the direction of the motion– the direction is along the surface (parallel to the surface)

• The magnitude of the kinetic frictional force depends only on the normal force and the properties of the 2 surfaces in contact

Where:– k is the coefficient of kinetic friction– FN is the normal force

Notes:– Kinetic friction is independent to the rate of travel of the sliding body– Kinetic friction is independent to the surface area of contact

k k Nf =μ F

kf

41

Tension Force

• Force applied through a rope or cable• When the rope or cable is mass-less (negligible

compared to the bodies it is attached to) it can be treated as a connection between 2 bodies– No mass means no force needed to accelerate rope– Force of pull transfers unchanged along the rope – Action force at one end is the same as the Reaction

force at the other end

• When attached to a pulley the tension force can be used to change the direction of force acting on a body

• Calculation of a tension force is usually an intermediate step to connecting the free-body diagrams between 2 attached objects

42

Tension Applications

• With Pulley (flat surface):

• Inclined Plane:

• Atwood Machine:

M1

M2

M1

M2

M1 M2

43

Sir Isaac Newton (1642-1727)

• Greatest scientific mind of the late 17th century• Invented the “calculus” (independently but

simultaneously with Gottfried Wilhelm Leibnitz)

• Among his accomplishments included:– A Particle Theory Light & Optics– A Theory of Heat & Cooling

• Established 3 Laws of Motion• Proposed the Law of Universal Gravitation (to

settle a parlor bet proposed by Christopher Wren!!)– The competition was actually between Edward

Haley & Robert Hooke– This theory successfully explained planetary motion

and elliptical orbits

“If I have seen further it is by standing on the shoulders of giants.”

44

General Physics I

Chapter 5: Uniform Circular Motion

Lecture Notes

45

Uniform Circular Motion

UCM, a special case of 2-D motion where:• The radius of motion (r) is constant• The speed (v) is constant • The velocity changes as object continually changes

direction• Since is changing (v is not but direction is), the object

must be accelerated:

• The direction of the acceleration vector is always toward the center of the circular motion and is referred to as the centripetal acceleration

• The magnitude of centripetal acceleration:

ˆ ˆyxc x y

vva =a a x + y

t t

vr

2

c c

va =a =

r

v

46

Derivation of Centrifugal AccelerationConsider an object in uniform circular motion, where:

r = constant {radius of travel}v = constant {speed of object}

The magnitude of the centripetal acceleration can be obtained by comparing the similar triangles for position and velocity:

x

v

r

y

x

y

The direction of position and velocity vectors both shift by the same angle, resulting in the following relation:

2

c

r v v r v v r= v= =

v a =

r v r r t rt

r1

r2

r

Position only

v1v2

v

Velocity onlyr2

v2

time, t2

r1

v1

time, t1

47

Centripetal Force

• Centripetal force is the center-seeking force that pulls an object into circular motion (object is not in equilibrium)

• The direction of the centripetal force is the same as the centripetal acceleration

• The magnitude of centripetal force is given by:

• For uniform circular motion, the centripetal force is the net force:

Notes:– the faster the speed the greater the centripetal force

– the greater the radius of motion the smaller the force

c cF = ma

2

c c c c

mvF = F = ma F = F = ma =

r

2

c c c

mvF = F = ma =

r

48

Centripetal Force & Gravitation

Assume the Earth travels about the Sun in UCM:

Questions:1. What force is responsible for maintain the Earth’s circular orbit?

2. Draw a Free-Body Diagram for the Earth in orbit.

3. Calculate gravitational force exerted on the Earth due to the Sun.

4. What is the centripetal acceleration of the Earth?

5. What is the Earth’s tangential velocity in its orbit around the Sun?

Sun

Earth

49

Centripetal Force vs. Centrifugal Force• An object moving in uniform circular motion experiences an

outward “force” called centrifugal force (due to the object’s inertia)

• However, to refer to this as a force is technically incorrect. More accurately, centrifugal force _______.– is a consequence of the observer located in an accelerated

(revolving) reference frame and is therefore a fictitious force (i.e. it is not really a force it is only perceived as one)

– is the perceived response of the object’s inertia resisting the circular motion (& its rotating environment)

– has a magnitude equal to the centripetal force acting on the body

Centripetal force & centrifugal force are NOT action-reaction pairs…

50

Artificial Gravity• Objects in a rotating environment “feel” an

outward centripetal force (similar to gravity)• What the object really experiences is the

normal force of the support surface that provides centripetal force

• The magnitude of “effective” gravitational acceleration is given by:

• If a complete rotation occurs in a time, T, then the rotational (or angular) speed is

Note: there are 2 radians (360o) in a complete rotation/revolution• Since v = r (rotational speed x radius), the artificial gravity is:

• Thus, the (artificial) gravitational strength is related to size of the structure (r), rotational rate () and square of the period of rotation (T2)

2

c artificial

va = g =

r

{where T is the period of the motion}2

= = T t

22

artificial 2

4 rg = ω r =

T

51

Banked Curves

• Engineers design banked curves to increase the speed certain corners can be navigated

• Banking corners decreases the dependence of static friction (between tires & road) necessary for turning the vehicle into a corner.

• Banked corners can improve safety particularly in inclement weather (icy or wet roads) where static friction can be compromised

• How does banking work?

• The normal force (FN) of the bank on the car assists it around the corner!

52

Vertical Circular Motion• Consider the forces acting on a motorcycle performing a

loop-to-loop:

• Can you think of other objects that undergo similar motions?

53

General Physics I

Chapter 6: Work & Energy

Lecture Notes

54

What is Energy?

• Energy is a scalar quantity associated with the state of an object (or system of objects).– Energy is a calculated value that appears in nature and whose total

quantity in a system always remains constant and accounted for.– Energy is said to be “Conserved”

• Loosely speaking, energy represents the “fuel” necessary for changes to occur in the universe and is often referred to as the capacity to perform work.

• We can think of energy as the “currency” associated with the “transactions” (forces!) that occur in nature.– In mechanical systems, energy is “spent” as force transactions are

conducted.– Alternatively, the exertion of force requires an “expenditure” of energy.

• The SI units for energy are called Joules (J)– In honor of James Prescott Joule

55

Work

• Work describes the energy transferred to/from an object by the exertion of a force

• Work is essentially a measure of useful physical output

Definition:standard form:

or

component form:

Notes:1. SI Units for Work: N.m

2. Unit comparison:

x y zW = F x + F y + F z

W = F cos = F s cos

s

F

s

F

s

1 J = 1 N mWork = Effort x Outcome

56

Two forces (a constant force and a non-constant force) move an object 1.5 m:

The area under the left graph is:Fconstant

.s = 5N x 1.5m = 7.5 N.m = W (work)

The area under the right graph is:Faverage

.s = 5N x 1.5m = 7.5 N.m = W (work)

Examples of Work(constant vs. variable force)

F

s

AREA

5 N

s = 1.5 m

Constant force

F

s

AREA

s = 1.5 m

5 NNon-Constant force

57

Kinetic Energy

The energy associated with an object’s state of motion– Kinetic energy is a scalar quantity that is never negative in value

Definition:

Key Notes:1. The kinetic energy for the x, y, and z components are additive

2. Kinetic energy is relative to the motion of observer’s reference frame, since speed and velocity are as well.

3. An object’s kinetic energy depends more on its speed than its mass

4. Any change in an object’s speed will affect a change in its kinetic energy

5. Unit comparison:

2 2 21 1 1x y z2 2 2

x y z

KE = mv + mv + mv

KE = KE + KE + KE

212KE = mv

2m

1 J = 1 kgs

2m

SI units: kgs

58

Work & Kinetic Energy• The net work performed on an object is related to the net

force:

• When , a change in state of motion & kinetic energy is implied:

Derivation:

NetF 0

Net NetW = F s cos = ma s cos

x yNet Net Net x yW = F x + F y= m a x + a y

Net fiW = KE - KE = KE

The Work-Energy Theorem!

fi

2 21x x x2from Ch 3: same for y

combining the abov a x = v -

e relati :v ons

fi fi

Net x y

2 2 2 21 1Net x x y y2 2

Net x y

Net

W = m a x + a y

W = m v -v + m v -v

W = KE + KE

W = KE

59

Work Performed by Gravitational Force

For a falling body (no air drag):

Gravitational force only performs work in the vertical direction:1. Wg is + when y is –

2. Wg is - when y is +

What about gravitational force on an incline?

gF =mg

y

y g

x g

g x y

W = F y = -mg y since cos = 1

W = F x = 0 since cos = 0

W = W + W = -mg y

gF =mg

y

60

Work Performed during Lifting & Lowering• Consider Joey “blasting” his pecs with a bench press

workout (assume vlift = constant).

Given:

Applying Newton’s 2nd Law:

The Work performed:

mg

y

LiftF

bar

bar

m = 100 kg

m g = -980 N

y= 1m

Net Lift bar bar

Lift bar

F F - m g = m a = 0

F = m g = 980 N

g bar

Lift Lift

Net Net Lift g

W = m g y = -980 N m

W = F y = 980 N m

W = F y = W + W = 0

61

Energy & the Body• Our bodies utilize energy to:

– Sustain cellular processes & maintain body temperature– Overcome joint friction– Produce motion

• In nutrition, energy is typically described in terms of Calories (i.e. the “calories” labeled on a cereal box):– These “nutritional” calories are actually kilocalories:

1 Calorie = 1 kcal = 1000 cal– The scientific calorie is related to the SI unit of energy:

1 cal = 4.186 J

Question(s): The “average” person requires ~2000 kcal per day. 1. How many joules are in 2000 kcal?2. How high could you climb with this amount of energy?3. It turns out the body uses only ~10% of its consumed energy for

physical activity. How high could you actually climb after consuming 2000 kcal?

62

SI units: The Watt (1 W = 1 J/s)

Note: Power is also related to Force & Velocity:

Power

• Power is a measure of work effectiveness

• Power is the time rate of energy transfer (work) due to an exerted force:Average Power:

Average Net Power:

dW F drP=

dt dt

avg

WP =

t

avgavg avg

F sWP = = = F v

t t

NetNet

W KEP = =

t t

63

Power (cont.)The same work output can be performed at various power

rates.

Example 1: Consider 100 J of work output accomplished over 2 different time intervals:

100 J over 1 s: 100 J over 100 s:

Example 2a: An 900 kg automobile accelerates from 0 to 30 m/s in 5.8 s. What is the average net power?

Example 2b: At the 30 m/s, how much force does the road exert on this vehicle? Use the same power as 2a.

100 JP= 100 W

1 s

100 JP= 1 W

100 s

.

2 21 m m2 s s 4o

net

900 kg 30 0K-KP = 6 98 10 W or 93.6 hp

t 5.8 s

..

44

ms

P 6 98 10 WP=F v 6 98 10 W F 2330 N

v 30

64

Conservative vs. Nonconservative Forces

Conservative Forces:1. When the configuration of a system is altered, a force performs work

(W1). Reversing the configuration of the system results in the force performing work (W2). The force is conservative if: W1= -W2

2. A force that performs work independent of the path taken.

3. A force in which the net work it performs around a closed path is always zero or:

Wnet = 0 J {for closed path}Examples:

Gravitational force (Fg), Elastic force (Fspring), and Electric force (FE)

Nonconservative Forces:1. The work performed by the force depends on the path taken

2. When the configuration of a system is altered then reversed, the net work performed by the force is not zero: Wnet ≠ 0 J {for closed path}

3. Work performed results in energy transformed to thermal energyExamples:

Air Drag (FDrag) and Kinetic friction (fkinetic)

65

Example: With Kinetic Friction (nonconservative force)

A 1 kg object (vo=5 m/s) travels up a 30o incline and back down against a 1.7 N

kinetic friction force. Note: Block will not travel up as far as previous example.

The Wup performed by fk (up):

The Wdown performed by fk (down):

The total Wnet performed:

Sliding on an InclineExample: No friction (conservative force)

A 1 kg object (vo=5 m/s) travels up a 30o incline and back down.

1. The Wnet performed by Fg (up):

2. The Wnet performed by Fg (down):

3. The total Wnet performed:

.2 21 m m

up 2 s sW = 1 kg 0 - 5 -12 5 J

.2 21 m m

down 2 s sW = 1 kg -5 - 0 12 5 J

gnet by FW -12.5 J + 12.5 J = 0 J

.f up kW = f x = -1.70 N 1 89 m = -3.21 J

.f down kW = f x = 1.70 N -1 89 m = -3.21 J

net by ff up f downW = W + W = - 3.21 J - 3.21 J = -6.42 J

66

Defining or Identifying a System• A system is a defined object (or group of objects) that are considered distinct from

the rest of its environment

• For a defined system:1. all forces associated strictly with objects within the defined system are deemed internal

forces• Internal forces do not transfer energy into/out of the system when performing work

within the system

Example: The attractive forces that hold the atoms of a ball together. These forces are ignored when applying Newton’s 2nd Law to the ball.

2. all forces exerted from outside the defined system are deemed external forces• External forces transfer energy into/out of the system when performing work on a

system

Example: The gravitational force that performs work on a falling object (the system) increases the ball system’s (kinetic) energy.

Note: When the ball and the earth are together defined as the system, the work performed by the gravitational force on the ball does NOT transfer energy into the system.

• The appropriate of a system determines when a force is considered internal or external & can go a long way toward simplifying the analysis of a physics problem

• The total energy associated with a defined system:

system thermal InternalE = PE + KE + E + E

67

Work Done on a System by External Forces

• For a defined system, external forces are forces that are not defined within the system yet perform work upon the system

• External forces transfer energy into or out of a system:

– Conservative external forces alter the U and K (a.k.a. the mechanical energy) of a system:

– Nonconservative external forces may alter the mechanical energy (U, K) as well as the non-mechanical energy (Einternal and/or Ethermal ) of a system:

NCExt

Internal therma

conExt Ext system

Ex lt

W = W + = E

W

W

E + = PE + KE + E

conExt systemW = E = PE+ KE

systemNCExt Internal thermal= E = PE+ KE+ W E + E

68

Conservation of Energy

• In general, the total energy associated with a system of objects represents the complete state of the system:

• Work represents the transfer of energy into/out of a system:

• For an isolated system, the total energy within a system remains a constant value:

or, for any 2 moments:

• Considering only the mechanical energy of the system:

InT tot ernal thermalE = PE + KE + E + E

Internal thermasystem lW = E = PE + KE + E + E

Internal tsyste h m lm er aE +E = PE + KE + = con E stant

Internal 1 thermasystem 1 1 2l 1 Internal 2 ther2 mal 2E = PE + KE + = PE + KE +E + E E + E

mech 1 1 2 2E = PE + KE = PE + KE KE = - PE Conservation of Mechanical Energy!

69

Deeper Thoughts on Cons. of Energy• Physicists have identified by experiment 3 fundamental

conservation laws associated with isolated systems:1. Conservation of Energy2. Conservation of Mass3. Conservation of Electric Charge

• Treated as accepted “facts”, these laws have allowed for experimental predictions that would not have been foreseen otherwise:1. Conservation of Energy led to the discovery of the neutrino during

neutron decay within the atomic nucleus2. Conservation of Mass is fundamental in the prediction of new

substance formed during chemical processes3. Conservation of Electric Charge predicts the formation of neutrons

do to the collision of protons with electrons, a process called Electron Capture.

• Considered as accepted “facts”, these laws have allowed for experimental predictions that would not have been foreseen otherwise.

70

Feynman on Energy

"There is a fact, or if you wish, a law, governing natural phenomena that are known to date. There is no known exception to this law - it is exact so far we know. The law is called conservation of energy [it states that there is a certain quantity, which we call energy that does not change in manifold changes which nature undergoes]. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity, which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same...”

Richard Feynman (1918-1988)

71

James Prescott Joule (1818-1889)

• English inventor & scientist• Interested in the efficiency of electric motors• Described the heat dissipated across a

resistor in electrical circuits (now known as Joules’ Law)

• Demonstrated that heat is produced by the motion of atoms and/or molecules

• Credited with establishing the mechanical energy equivalent of heat

• Participated in establishing the “Law of Energy Conservation”

"It is evident that an acquaintance with natural laws means no less than an acquaintance with the mind of God therein expressed."

72

General Physics I

Chapter 7: Momentum & Impulse

Lecture Notes

73

Linear Momentum

Linear momentum ( ) represents inertia in motion (Newton described momentum as the “quantity of motion”)

Conceptually: reflects the effort required to bring a moving object to rest depends not only on its mass (inertia) but also on how fast it is moving

Definition: • Momentum is a vector quantity with the same direction as

the object’s velocity• SI units are kg.m/sNewton’s 1st Law revisited:

The momentum of an object will remain constant unless it is acted upon by a net force (or impulse)

p = mv

p

74

Impulse-Momentum TheoremIn general, Newton’s 2nd Law, can be rewritten as

Rearranging terms:

this is called the Net (or average) Impulse!!Definition of Impulse associated with an applied force:

• The SI units for impulse are N.s

• Impulse represents simultaneously:1. The product of the force times the time:

2. The change in linear momentum of the object:

this is the simple casenet net where m is constant!

mv p vF = = Note: F = m = ma

t t t

net net avgp = F t when F is not constant: p = F t

avgJ = p = F t

avgJ = F t

fiJ = p = mv - mv

75

Notes on Impulse• Impulses always occur as action-reaction pairs (according to Newton’s

3rd Law)• The force.time relationship is observed in many “real world” examples:

– Automobile safety:• Dashboards• Airbags• Crumple zones

– Product packaging• Styrofoam spacers

– Sports• Tennis: racket string tension• Baseball: “juiced” baseballs & baseball bats (corked & aluminum vs. wood)• Golf: the “spring-like” effect of golf club heads• Boxing gloves: (lower impulsive forces in the hands)

76

A Superman ProblemIt is well known that bullets and missiles bounce off Superman’s chest. Suppose a bad guy sprays Superman’s chest with 0.003 kg bullets traveling at a speed of 300 m/s (fired from a machine gun at a rate of 100 rounds/min). Each bullet bounces straight back with no loss in speed. Problems:a) What is the impulse exerted on Superman’s

chest by a single bullet?

b) What is the average force exerted by the stream of bullets on Superman’s chest?

                              

 

77

CollisionsA specific type of interaction between 2 objects. The basic assumptions of a collision:

1. Interaction is short lived compared to the time of observation2. A relatively large force acts on each colliding object3. The motion of one or both objects changes abruptly following

collision4. There is a clean separation between the state of the objects

before collision vs. after collision

3 classifications for collisions:– Perfectly elastic: colliding objects bounce off each other and no

energy is lost due to heat formation or deformation (Ksystem is conserved)

– Perfectly inelastic: colliding objects stick together (Ksystem is not conserved)

– Somewhat inelastic (basically all other type of collisions): KE is not conserved

1 21ip

2ip 1 2

1fp

2fp

1 21 21ip

2ip

1f+2fp

78

Conservation of Linear Momentum

The total linear momentum of a system will remain constant when no external net force acts upon the system, or

• Note: Individual momentum vectors may change due to collisions, etc. but the linear momentum for the system remains constant

• Useful for solving collision problems:– Where all information is not known/given– To simplify the problem

• Conservation of Momentum is even more fundamental than Newton’s Laws!!

1 2 before collision 1 2 after collision(p + p + ...) = (p + p + ...)

79

Conservation of Momentum (Examples)• The ballistic pendulum• 2 body collisions (we can’t solve 3-body systems…)

– Perfectly inelastic (Epre-collision ≠ Epost-collision)

– Perfectly elastic (Epre-collision = Epost-collision)

• Collisions in 2-D or 3-D:– Linear momentum is conserved by components:

By Components:

ˆ ˆ1x 2x 1x 2x

before collision after collision(p + p + ...) i = (p + p + ...) i

1 2 before collision 1 2 after collision(p + p + ...) = (p + p + ...)

ˆ ˆ1y 2y 1y 2y

before collision after collision(p + p + ...) j = (p + p + ...)j

80

Notes on Collisions & Force• During collisions, the forces generated:

– Are short in duration– Are called impulsive forces (or impact forces or collision forces)

– Often vary in intensity/magnitude during the event– Can be described by an average collision force:

Example: a golf club collides with a 0.1 kg golf ball (initially at rest), t 0.01s. The velocity of the ball following the impact is 25 m/s.

The impulse exerted on the ball is:

The average impulsive force exerted on the ball is:

The average impulsive force exerted on the club is:

. .Net avg

p impulseF = F =

t timei e

ˆ ˆm ms sp = m v = (0.1 kg)(25 - 0 ) i = 2.5 N s i

ˆ ˆavg

p 2.5 N sF = = i = 250 N i

t 0.01 s

ˆ ˆavg

p -2.5 N sF = = i = -250 N i

t 0.01 s

81

Center of MassCenter of Mass ( ) refers to the average location of mass for a

defined mass.• To determine the center of mass, take the sum of each mass

multiplied by its position vector and divide by the total mass of the system or

• Note, if the objects in the system are in motion, the velocity of the system (center of mass) is:

• When psystem = 0 (i.e. Fext = 0) then vcm = constant– The motion of all bodies even if they are changing individually will

always have values such that vcm = constant

cmr

n

i i1 1 2 2 3 3 n n i=1

cm1 2 3 n sys

mrm r + m r + m r + ... + m r

r = = m + m + m + ... + m m

n

i i1 1 2 2 n n i=1

cm1 2 n sys

mvm v + m v + ... + m v

v = = m m +...+ m m

82

Rene Descartes (1596-1650)

• Prominent French mathematician & philosopher

• Active toward end of Galileo’s career• Studied the nature of collisions

between objects• First introduced the concept of

momentum (called it “vis-à-vis”) – he defined “vis-à-vis” as the product of

weight times speed• Demonstrated the Law of Conservation

of Momentum

“Each problem that I solved became a rule which served

afterwards to solve other problems.”

83

General Physics I

Chapter 8: Rotational Kinematics

Lecture Notes

84

Rotational Motion & Angular Displacement

• When an object moves in a circular path (or rotates):– It remains a fixed distance (r) from the center of the circular path (or

axis of rotation)– Since radial distance is fixed, position can be described by its angular

position ()• Angular position () describes the position of an object along

a circular path– Measured in radians (or degrees)

• Angular displacement:

• Angular velocity: the rate at which angular position changes:

• Angular acceleration: is the rate at which angular velocity changes:

θω =

t

fi iθ = θ - θ = θ when θ is 0

ω =

t

85

Relationship between rotational & linear variables for circular motion

Position:Where

Displacement (arc length): s = r

Linear (tangential) speed: vT = r

Linear Acceleration: a = r

r

R

When to = 0 and is constant:

1 - o = t

2 - o = ½(o + )t

3 - o = o + ½ t2

4 2 - o2 = 2

Equations of Rotational Kinematics

R = θ r

ˆ ˆand x = r sinθ x y = r cosθ y

s

Ri

Rf

86

Centripetal & Tangential AccelerationFor an object moving in uniform circular motion, the magnitude of its centripetal acceleration is:

Since v = r v2 = (r)2 therefore:

ac = 2rWhen is not constant, the effect of angular acceleration must also included:

Or

Thus, an object’s circular motion can be generalized in terms of and Notes:1. The 2 components of acceleration are perpendicular to each other2. To determine the magnitude & direction of a, they must be treated as vectors

ˆ ˆc a = -a r + r r

ˆ ˆ2 a = -ω r r + r r

2

c

va =

r ac

v

r

ac

r

r

87

General Physics I

Chapter 9: Rotational Dynamics

Lecture Notes

88

Rigid Objects & TorqueTorque is a vector quantity that represents the application of

force to a body resulting in rotation (or change in rotational state)

Definition:• The SI units for torque are N.m (not to be confused with joules)

• Torque depends on:– The Lever arm (leverage), – The component of force perpendicular to lever arm,

r

= F r = F sin r

ˆF = F sin i

F

r

r

F

89

Newton’s 2nd Law(for Rotational Motion)

When there is a net torque exerted on a rigid body its state of rotation ( ) will change depending on:1. Amount of net torque,2. Rotational inertia of object, I

Note: No net torque means = 0 and = constant}

Alternatively, the net torque exerted on a body is equal to the product of the rotational inertia and the angular acceleration:

This is Newton’s 2nd Law (for rotation)!!n

1 2 ii=1

= + + ... = = I

n1 2

ii=1

+ + ... 1 = = =

I I I

ω

90

Rigid Objects in EquilibriumA rigid body is in equilibrium when:

– No net force no change in state of motion (v = constant)

– No net torque no change in state of rotation(= constant)

Both of the above conditions must be met for an object to be in mechanical equilibrium!

Note: an object can be both moving and rotating and still be in mechanical equilibrium

x y zF = F + F + F = 0

= 0

91

Center of Gravity (COG)The COG is the location where all of an

object’s weight can be considered to act when calculating torque

• The COG may or may not actually be on the object (i.e. consider a hollow sphere or a ring)

• When an object’s weight produces a torque on itself, it acts at its center of mass

To calculate center of gravity: n

i i1 1 2 2 i=1

cg1 2 tot

WxW x + W x + ...

x = = W + W + ... W

n

i i1 1 2 2 i=1

cg1 2 tot

WyW y + W y + ...

y = = W + W + ... W

92

Moment of InertiaThe resistance of a rigid body to changes in its state of rotational motion ( ) is called the moment of inertia (or rotational inertia)

The Moment of Inertia (I) depends on:1. Mass of the object, m2. The axis of rotation3. Distribution (position) of mass about the axis of rotation

Definition (For a discrete distribution of mass):

Where:mi is the mass of a small segment of the objectri is the distance of the mass mi from the axis of rotation

The SI units for I are kg.m2

Note: For a continuous distribution of mass there are more sophisticated techniques for calculating the moment of inertia

n2 2 2 2

1 1 2 2 n n i ii=1

I = m r + m r + ... m r = mr

ω

93

Moment of Inertia (common examples)1. A simple pendulum:

2. A thin ring:

3. A solid cylinder:

2pendulumI = mR

m

R

2ringI = mR

dm

R

2

cylinder

mRI =

2

H

R

dm

94

Moment of Inertia (common examples, cont.)4. A hollow sphere:

5. A solid sphere:

6. A thin rod:

. m2

rod

mLI =

12

mR

mR

2

hollow sphere

2mRI =

3

2

solid sphere

2mRI =

5

L

95

The Parallel Axis Theorem• The Parallel Axis Theorem is used to determine the moment

of inertia for a body rotated about an axis a distance, l, from the center of mass:

Example: A 0.2 kg cylinder (r=0.1 m) rotated about an axis located 0.5 m from its center:

2cmI = I + mL

L

2cm

22

2

2

I = I + mL

mrI = + mL

2I = 0.002 + 0.050 kg m

I = 0.052 kg m

96

Work & Rotational Kinetic EnergyWhen torque is applied to an object and a rotation is produced, the torque does work:

When there is net torque:

Since 2 - o2 = 2

The rotational kinetic energy (Krot) for an object is defined as:

This is the Work-Energy Theorem (for rotation)!!

W = θ

NetW = θ = I θ 2 2

2 2o 1 1Net o2 2

ω -ωW = I θ = I = Iω - Iω

2

o

2 2 21 1 1rot Net o rot rot rot2 2 2K = Iω W = Iω - Iω = K -K = K

97

Rolling objects & Inclined PlanesConsider a solid disc & a hoop rolling down a hill (inclined plane):

1. Apply Newton’s 2nd Law (force) to each object2. Apply Newton’s 2nd Law (torque) to each object3. What is the acceleration of each object as they roll down the

hill?4. Which one reaches the bottom first?

Solid disc

Hoop

98

Angular Momentum• Angular momentum is the rotational analog of linear momentum• It represents the “quantity of rotational motion” for an object (or its

inertia in rotation)• Angular Momentum (a vector we will treat as a scalar) is defined as:

L = I. • Note:

Angular Momentum is related to Linear Momentum:

L = r.p sin r is distance from the axis of rotationp is the linear momentum is the angle between r and p

• SI units of angular momentum are kg.m2/s r

p

99

Conservation of Angular Momentum• When there is no net torque acting on an object (or no net torque acts on

a system) its angular momentum (L) will remain constant or

L1 = L2 = … = a constant valueOr

I11 = I22 = etc…• This principle explains why planets move faster the closer the get to

the sun and slower the they move when they are farther away

Note: this is a re-statement of Newton’s 1st Law

slowest

fastest

100

Conservation of Angular Momentum (Examples)• A figure skater• A high diver• Water flowing down a drain

101

Archimedes (287-212 BC)

• Possibly the greatest mathematician in history

• Invented an early form of calculus• Discovered the Principle of Buoyancy (now

called Archimedes’ Principle)

• Introduced the Principle of Leverage (Torque) and built several machines based on it.

“Give me a point of support and I will move the Earth”