CH 01: INTRODUCTION & MATHEMATICAL CONCEPTSlightcat-files.s3.amazonaws.com/packets/admin_physics-3... · 2019-11-04 · We measure _____ in nature. Measurements must have ... CH 01:

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PHYSICS - CUTNELL 11E

CH 01: INTRODUCTION & MATHEMATICAL CONCEPTS

● Physics is the study of natural phenomena, including LOTS of measurements and equations. Physics = math + rules.

UNITS IN PHYSICS

● We measure __________________________ in nature. Measurements must have ____________.

- For equations to “work”, units have to be ___________________ with one another. For example:

F = m a ( Force = Mass * Acceleration )

[ ] = [ ] [ ]

● A group of units that “work together” is called a ____________ of units.

- In Physics, we use the _____________ ( Système International ).

- Many other conventions / standards are used in Physics.

● Non-S.I. measurements must be converted into S.I. before we plug them into equations. VIDEO: Unit Conversions

- Equations must be dimensionally consistent: units are the same on both sides. VIDEO: Unit Analysis*

PRECISION, SIGNIFICANT FIGURES, AND SCIENTIFIC NOTATION

● Measurements must be precise. Precision has to do with how many _________________________ a measurement has.

- When adding/multiplying/etc. measurements, we must use Sig Fig Rules*. VIDEO: Significant Figures*

- To “compress” numbers: (a) use powers of TEN; (b) round to TWO* decimal points (digits 5 or greater, round up).

- We can use _______________________ to represent numbers ____________.

EXAMPLE 1: For each number below: (a) How many sig figs does it have? (b) Re-write it in scientific notation (2 decimals).

(a) 010,000 (d) 00.010

(b) 010,000.0 (e) 382,400

(c) 010,010 (f) 23,560.0

QUANTITY S.I. UNIT

[ ]

[ ]

[ ]



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CONVERTING UNITS

● If given units on different systems, convert all units to the same system. Usually this means converting to S.I. units.

EXAMPLE 1: Convert 44 lb into kg (1 kg = 2.2 lb):

PATTERN: [ ] [ ] (

) (

) = [ ] [ ]

____________ _____________________ _____________

EXAMPLE 2: Convert 100 m2 into ft2 (1 ft = 0.305 m):

[ ] [ ] (

) (

) = [ ] [ ]

EXAMPLE 3: Convert 30 m/s into miles per hour (1 mi = 1600 m, 1 hour = 3600 s):

[ ] [ ] (

) (

) = [ ] [ ]

PRACTICE 4: Convert 10 ft into km (1 ft = 0.305 m):

(a) 0.00305 km

(b) 327.87 km

(c) 0.0305 km

(d) 30.5 km

PRACTICE 5: Convert 2,000 mL into m3 (1 L = 1,000 cm3):

(a) 0.002 m3

(b) 0.2 m3

(c) 20 m3

(d) 2,000 m3

PRACTICE 6: Convert 80 km/h into m/s (1 km = 1000 m, 1 hour = 3600 s):

(a) 288 m/s

(b) 28.8 m/s

(c) 4.8 m/s

(d) 22.22 m/s



Page 3

UNIT ANALYSIS (aka Dimensional Analysis)

● Equations must be dimensionally consistent: Units on both sides of the equation must be the same.

- For example, from speed =

we can get that distance = speed * time

- Because distance is measured in meters, and time in seconds: speed =

- The second equation is also dimensionally consistent: distance = speed * time =

EXAMPLE 1: Suppose Wikipedia says that the distance Y

(in meters) that an object free falls from rest in t seconds is

given by Y = ½ g t2, where g is the acceleration due to

gravity, in m/s2. Is this equation dimensionally consistent?

EXAMPLE 2: Hooke’s law states that the restoring force F

on a spring is related to its displacement from equilibrium x

by F = – k x. If F is measured in Newtons (N) and x in

meters (m), what unit(s) must the force constant k have?

PRACTICE 3: If distance x is measured in [m], time t in [s],

speed v in [m/s], acceleration a in [m/s2], mass m in [kg],

and force F in [N] = [kg * m/s2], show which of the following

equations is/are dimensionally consistent:

(a) v2 = 2 a x

(b) v2 = 2 a t

(c) F = m (v2 – v1) / t

PRACTICE 4: Newton’s Law of Gravitation states that the

attraction between two objects is given by F = G m1 m2 / r2,

where F is in [kg * m/s2], m1 and m2 are the objects’ masses

(in [kg]), and r is the distance between them (in [m]). What

unit(s) must the Universal Constant G have?

(a) kg s2 / m3

(b) m3 / kg s2

(c) m / s2

(d) m3 / s2



Page 4

CONCEPT: INTRODUCTION TO VECTORS AND SCALARS ● When we take measurements, you always get the ______________ (size of measurement). [ Example: 60°F, 10kg ]

- SOME measurements also have ______________. [ Example: 10m right, 20 miles/hr NORTH ]

- Measurements with direction are [ Vectors | Scalars ]; measurements without direction are [ Vectors | Scalars ]

Measurement Quantity Magnitude? Direction? Vector/Scalar

“Apple weighs 5kg” Mass [ Vector | Scalar ]

“Days are 24hr long” Time [ Vector | Scalar ]

“It’s 60°F outside” Temperature [ Vector | Scalar ]

“I pushed with 100N left” Force [ Vector | Scalar ]

“I walked for 10 ft” [ Vector | Scalar ]

“I walked 10 ft. east” [ Vector | Scalar ]

“I drove at 80 mph” [ Vector | Scalar ]

“I drove 80mph west” [ Vector | Scalar ]



Page 5

CONCEPT: DISPLACEMENT VS. DISTANCE ● There are two similar-sounding words to measure how FAR something moves (Length):

EXAMPLE: Find the displacement and total distance traveled from A to B for each of the following situations:

● Displacements can sometimes be negative, but distances are ALWAYS positive.

- In Physics, (+ / -) signs are usually used to indicate direction!

DISTANCE (𝒅) DISPLACEMENT (𝜟�⃗⃗� )

A B

𝒙𝟎 = −𝟐 𝒙 = 𝟕

A B

𝒙 = 𝟑 𝒙𝟎 = 𝟕

A B

𝒙 = 𝟒 𝒙 = 𝟏𝟎

_____ = ___________ _____ = ___________

𝑑 = ______________ → Scalar (Magnitude only)

- ____________________ between initial & final position

Δ𝑥 = ______________ → Vector (Mag. + Dir.)

- _______________ of all lengths traveled - ______________ in position (___)

10m

6m

10m

6m



Page 6

PRACTICE: Starting from a pillar, you run 140m east (the +x-direction), then turn around. (a) How far west would you have to walk so that your total distance traveled is 300m? (b) What is the magnitude and direction of your total displacement?



Page 7

CONCEPT: INTRO TO VECTOR MATH

● Adding/subtracting scalars is easy. But vectors have direction, so math with vectors is sometimes not as straightforward.

- Because vectors have direction, they’re drawn as ________________.

EXAMPLE: For each of the following situations, draw your displacement vectors and calculate the total displacement:

COMBINING

SCALARS

3 kg 4 kg

“You combine a 3kg & 4kg box”

COMBINING PERPENDICULAR

VECTORS

Total Mass: 3kg + 4kg = _______ Total Displacement: __________

“You walk 3m right, then 4m up”

● Forms _____________.

- just TRIANGLE MATH

● Simple Addition

(a) You walk 10m to the right, and then 6m to the left (b) You walk 6m to the right, and then 8m down

COMBINING PARALLEL VECTORS

“You walk 3m right, then 4m right”

Total Displacement: __________

● Add just like normal numbers



Page 8

PRACTICE: Two perpendicular forces act on a box, one pushing to the right and one pushing up. An instrument tells you

the magnitude of the total force is 13N. You measure the force pushing to the right is 12N. Calculate the force pushing up.



Page 9

CONCEPT: ADDING VECTORS GRAPHICALLY

● Vectors are drawn as arrows and are added by ______________ the arrows (tip-to-tail).

● The RESULTANT vector (�⃗⃗� or �⃗⃗� ) is always the SHORTEST PATH from the start of the first vector → end of the last.

- Adding vectors does NOT depend on the order (commutative), so �⃗⃗� + �⃗⃗� = �⃗⃗� + �⃗⃗� .

EXAMPLE: Find the magnitude of the Resultant Vector �⃗⃗� = �⃗⃗� + �⃗⃗� .

⇔

ADDING PERPENDICULAR VECTORS

3m

4m

ADDING ANY VECTORS

�⃗⃗�

�⃗⃗�

�⃗⃗� + �⃗⃗� �⃗⃗� + �⃗⃗�

𝒚

𝒙

�⃗⃗�

�⃗⃗�

Resultant Vector: (Total Displacement)

____________


____________


____________



Page 10

PRACTICE: A delivery truck travels 8 miles in the +x-direction, 5 miles in the +y-direction, and 4 miles again in the

+x-direction. What is the magnitude (in miles) of its final displacement from the origin?

𝒚

𝒙



Page 11

EXAMPLE: Find the magnitude of the Resultant Vector �⃗⃗� = �⃗⃗� + �⃗⃗� + �⃗⃗� .

𝒚

𝒙 −𝒙

−𝒚

�⃗⃗�

�⃗⃗�

�⃗⃗�



Page 12

CONCEPT: SUBTRACTING VECTORS GRAPHICALLY

● Subtracting vectors is exactly like adding vectors tip-to-tail, but one (or more) of the vectors gets _______________.

EXAMPLE: Find the magnitude of the Resultant Vector �⃗⃗� = �⃗⃗� − �⃗⃗� .

⇔

ADDING VECTORS

�⃗⃗� − �⃗⃗� �⃗⃗� − �⃗⃗�

SUBTRACTING VECTORS

�⃗⃗� + �⃗⃗� �⃗⃗� + �⃗⃗�

● When adding, order [ DOES | DOES NOT ] matter

● “Negative” vector: SAME magnitude, ____________ direction

● When subtracting, order [ DOES | DOES NOT ] matter

𝒚

𝒙

𝒚

𝒙

𝒚

𝒙

�⃗⃗�

�⃗⃗�

𝒚

𝒙

Resultant → shortest path: (Total Displacement)

_______________


_______________


_______________

�⃗⃗�

�⃗⃗�

𝒚

𝒙

�⃗⃗�

�⃗⃗�



Page 13

PRACTICE: Find the magnitude of the Resultant Vector �⃗⃗� = �⃗⃗� − �⃗⃗� − �⃗⃗� .

𝒚

𝒙 −𝒙

−𝒚

�⃗⃗� �⃗⃗�

�⃗⃗�



Page 14

CONCEPT: ADDING MULTIPLES OF VECTORS

● When you multiply a vector by a number (𝐴 → 2𝐴 ), the magnitude (length) changes but NOT the direction.

EXAMPLE: Find the magnitude of the Resultant Vector �⃗⃗� = 𝟑�⃗⃗� − 𝟐�⃗⃗� .

𝒚

𝒙

ADDING VECTORS

ADDING MULTIPLES OF VECTORS

�⃗⃗�

�⃗⃗�

● Multiplying by > 1 [ increases | decreases ] magnitude/length

● Multiplying by < 1 [ increases | decreases ] magnitude/length

�⃗⃗� + �⃗⃗� 𝟐�⃗⃗� + 𝟎. 𝟓�⃗⃗� 𝒚

𝒙

Resultant Vector → Shortest Path: (Total Displacement)

_______________

𝒚

𝒙

�⃗⃗� �⃗⃗�

Resultant Vector → Shortest Path: (Total Displacement)

_______________



Page 15

CONCEPT: VECTOR COMPOSITION AND DECOMPOSITION

● You’ll need to do vector math without using grids/ squares.

- Vectors have magnitude (length), direction (angle 𝜽𝒙), and components (legs).

EXAMPLE: For each of the following, draw the vector and solve for the missing variable(s).

VECTOR DECOMPOSITION

𝑨𝒙 = __________

𝑨𝒚 = __________

VECTOR COMPOSITION

𝑨 = √𝑨𝒙 𝟐 + 𝑨𝒚

𝟐

𝜽𝒙 = ____________

VECTOR COMPOSITION VECTOR DECOMPOSITION

● Use SOH-CAH-TOA to decompose �⃗⃗� →components 𝐴𝑥 & 𝐴𝑦.

- Angle 𝜽𝒙 must be drawn to nearest ________

1D Components → 2D Vector (Magnitude & Direction)

● Components 𝑨𝒙 & 𝑨𝒚 combine → magnitude �⃗⃗�

- Points in direction 𝜽𝒙

a) Ax = 8m, Ay = 6m, 𝑨 = ? θx = ? b) B = 13m, θx = 67.4°, Bx = ? By = ?

+𝒚

+𝒙 3

4

2D Vector (Magnitude & Direction) → 1D Components

θx=53°

5

+𝒚

+𝒙

𝒚

𝒙

𝒚

𝒙



Page 16

EXAMPLE: A vector A has y-component of 12 m makes an angle of 67.4° with the positive x-axis. (a) Find the magnitude of

A. (b) Find the x-component of the vector.

Vector Composition

(Components→Vector)

Vector Decomposition

(Vector→Components)

𝑨 = √𝑨𝒙 𝟐 + 𝑨𝒚

𝟐

𝜽𝑿 = 𝐭𝐚𝐧−𝟏 (𝑨𝒚

𝑨𝒙)

𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝑿)

𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝑿)



Page 17

CONCEPT: VECTOR ADDITION BY COMPONENTS

● You’ll need to add vectors together and calculate the magnitude & direction of the resultant without counting squares.

EXAMPLE: You walk 5m at 53° above the +x-axis, then 8m at 30° above the +x-axis. Calculate the magnitude & direction

of your total displacement.

VECTOR ADDITION

1) Draw & connect vectors tip-to-tail

2) Draw Resultant & components

3) Calculate ALL X&Y components

4) Combine X & Y components according to R equation

5) Calculate R and 𝜃𝑅

Vector Composition




𝑹 = √𝑹𝒙 𝟐 + 𝑹𝒚

𝟐

𝜽𝑿 = 𝐭𝐚𝐧−𝟏 (𝑹𝒚

𝑹𝒙)


𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝑿) x y

�⃗⃗�

�⃗⃗�

�⃗⃗� = ______

ADDING VECTORS GRAPHICALLY (WITH SQUARES)

ADDING VECTORS BY COMPONENTS (WITHOUT SQUARES)

+𝒚

+𝒙

+𝒚

+𝒙

�⃗⃗�

�⃗⃗�



Page 18

EXAMPLE: Vector �⃗⃗� has a magnitude of 10m at a direction 40° above the +x-axis. �⃗⃗� has magnitude 3 at a direction 20°

above the x-axis. Calculate the magnitude and direction of �⃗⃗� = �⃗⃗� − 𝟐�⃗⃗� .

VECTOR ADDITION

1) Draw & connect vectors tip-to-tail

2) Draw Resultant & components

3) Calculate ALL X&Y components

4) Combine X & Y components according to R equation

5) Calculate R and 𝜃𝑅

Vector Composition




𝑹 = √𝑹𝒙 𝟐 + 𝑹𝒚

𝟐

𝜽𝑿 = 𝐭𝐚𝐧−𝟏 (𝑹𝒚

𝑹𝒙)


𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝑿)

𝒚

𝒙



Page 19

CONCEPT: INTRO TO DOT PRODUCT (SCALAR PRODUCT)

● Multiplying Vectors by Scalars is simple. You’ll need to know 2 different ways to multiply Vectors by other Vectors:

1) Dot Product (Scalar Product): _______

2) Cross Product (Vector Product): _______ (covered later)

EXAMPLE: Calculate the Dot Product of �⃗⃗� and �⃗⃗� in each of the following:

+𝒚

+𝒙 −𝒙

a)

�⃗⃗� = 4 @ 0°

�⃗⃗� = 3 @ 0°

Multiples of Vectors Dot Product

4 times 3 =

4 ● =

3

Vector * Scalar (#) = Vector (number + direction) Vector • Vector = Scalar (number only, no direction)

�⃗⃗� ● �⃗⃗� = _____________ - 𝜽 = smallest angle from �⃗⃗� to �⃗⃗�

- Put calculator in degrees mode!

+𝒚

+𝒙 −𝒙

b)

+𝒚

+𝒙 −𝒙

c)

+𝒚

+𝒙 −𝒙

d)

+𝒚

+𝒙 −𝒙

e)

�⃗⃗� = 4 @ 60°

�⃗⃗� = 3 @ 0°

�⃗⃗� = 4 �⃗⃗� = 3

�⃗⃗� = 4

�⃗⃗� = 3

�⃗⃗� = 4

�⃗⃗� = 3

- Dot Product = multiplication of ____________ components.

- Negative Dot Product = components in ____________ directions.

60°

- ZERO Dot Product = components in ______________ directions.

- Always line up vectors end-to-end (tail-to-tail)



Page 20

PRACTICE: Using the vectors given in the figure, (a) find �⃗⃗� ● �⃗⃗� . (b) Find �⃗⃗� ● �⃗⃗� .

𝒚

𝒙

�⃗⃗� =15 �⃗⃗� =10

�⃗⃗� =20

30°

53°

30°



Page 21

CONCEPT: UNIT VECTORS

● Vectors are sometimes represented using a special notation called Unit Vectors.

● Unit vectors make vector addition very straightforward:

EXAMPLE: Vector �⃗⃗� = 4𝑖̂ + 2𝑗 ̂and �⃗⃗� = −�̂� + 𝟐𝒋.̂ Draw the vectors and calculate �⃗⃗� = �⃗⃗� + �⃗⃗� in unit vector form.

Vector Addition w/ Unit Vectors

�⃗⃗� = 𝐴𝑥 �̂� + 𝐴𝑦𝒋̂ = ______________

�⃗⃗� = 𝐵𝑥 �̂� + 𝐵𝑦𝒋̂ = ______________

�⃗⃗� = �⃗⃗� + �⃗⃗� = ____________________

�̂� points in ____ direction.

𝒋̂ points in ____ direction.

𝒌 points in ____ direction.

𝒚

𝒙

Graphical Magnitude & Direction Unit Vector

𝒚

𝒙

“5m @ 53°” +𝒚

+𝒙

+𝒛

● special kind of vector that __________ in a direction

- has magnitude/length ____.

“ 3�̂� + 4𝒋̂”



Page 22

PRACTICE: �⃗⃗� = (4.0 m)�̂� + (3.0 m)𝒋̂ and �⃗⃗� = (−13.0 m)�̂� + (7.0 m)𝒋̂. You add them together to produce another vector �⃗⃗� .

(a) Express this new vector �⃗⃗� in unit-vector notation. (b) What are the magnitude and direction of �⃗⃗� ?



Page 23

EXAMPLE: Consider the three displacement vectors �⃗⃗� = (3 �̂� − 3 𝒋̂) m, �⃗⃗� = (�̂�̂ − 4 𝒋̂̂) m, and �⃗⃗� = (−𝟐 �̂� + 𝟓 𝒋̂) m.

(a) Find the magnitude and direction of D = A + B + C.

(b) Find the magnitude and direction of E = −A − B + C.



Page 24

Documents

CH 01: INTRODUCTION & MATHEMATICAL CONCEPTSlightcat-files.s3.amazonaws.com/packets/admin_physics-3... · 2019-11-04 · We measure _____ in nature. Measurements must have ... CH 01: