Chapter 1 Outline Units, Physical Quantities, and Vectors Idealized models Units SI units, prefixes, and unit consistency Uncertainty and significant figures Order of magnitude approximations Vectors and scalars Component notation Vector addition and subtraction Dot and cross products

Physics Physics is an experimental science. Observation leads to theory. Theories have limits, or ranges of validity. Solving problems – Idealized models We make approximations in order to solve problems; you wouldn’t use general relativity to solve the problem of a body in free-fall! Always keep in mind what simplifications are inherent in your model, and think about whether they are reasonable.

Standards and Units We need to describe most physical phenomena qualitatively, so we compare our measurement of a physical quantity to some standard reference, or unit. The standard system of units is the International System, or SI. In the United States, we often use US (or British) customary units (inches, pounds…), but we will only use SI units in this course The SI base units (Other units are derived from these.) Length: meter (m) Mass: kilogram (kg) Time: second (s) Electric current: ampere (A) Thermodynamic temperature: kelvin (K) Amount of substance: mole (mol) Luminous intensity: candela (cd)

Unit Prefixes In order to introduce larger or smaller units, we use prefixes. These are some of the more common prefixes: Power of Ten Prefix Abbreviation 10 12 tera- T 10 9 giga- G 10 6 mega- M 10 3 kilo- k 10 −2 centi- C 10 −3 milli- m 10 −6 micro- μ 10 −9 nano- n 10 −12 pico- p 10 −15 femto- f

Unit Consistency All equations must be dimensionally consistent. Each side of the equation (or any terms that are added) must have the same units For example: A body moving at a constant speed 𝑣=2 m/s will travel a distance 𝑑=10 m in time 𝑡=5 s. 𝑑=𝑣𝑡 10 m= 2 m s 5 s=10 m s s=10 m Likewise, for unit conversions, you multiply by terms that are equal to one, such as 60 s 1 min . 60 mi/hr=60 mi hr 1609 m 1 mi 1 hr 3600 s =26.82 m/s

Uncertainty and Significant Figures All measurements have some level of uncertainty. We can express this uncertainty (or error) as a number plus/minus the uncertainty. For example, if the mass of a steel ball is given as 1.24±0.03 kg, then the ball is unlikely to be greater than 1.27 kg or less than 1.21 kg. This uncertainty can also be expressed as a fraction or percent of the given value. If the uncertainty is not explicitly stated, we can go by the number of significant figures (s.f.). We assume an uncertainty of one in the least significant digit. When multiplying and dividing, the answer has the same number of s.f. as the term with the fewest s.f. When adding, use the location of the decimal point.

Order of Magnitude Estimates It is important to develop a sense of what is reasonable to expect for an answer to a question. If you are calculating the speed of a pitched baseball, would 4 m/s be reasonable? 40 m/s? 400 m/s? Sometimes, you can catch errors in your solution by examining the plausibility of your calculated answer. Also, there are times when the data needed to do exact calculations are not available. In this case, we might make an order of magnitude estimate.

Order of Magnitude Estimates Example Problem 1.21 – How many times does a typical person blink their eyes in a lifetime?

Vectors and Scalars Some physical quantities are fully described by a single number with a unit, such as mass, length, time. These are scalar quantities, and only have a magnitude. We can use regular arithmetic to combine these quantities. Other physical quantities, such as velocity or force, must include a direction. These are vector quantities, and have both a magnitude and a direction. We must use vector arithmetic to combine these quantities.

Vector Notation First, some notation issues: Vectors are drawn as an arrow pointing in the vector’s direction and a length proportional to its magnitude. Multiplying a vector by −1 reverses the direction of the vector. Vectors are represented by a letter, generally in bold, with an arrow, such as 𝑨 . Unit vectors have a magnitude of one, and are therefore used to show direction. They are distinguished by a caret or “hat” instead of an arrow, e.g., 𝒙 . The magnitude of a vector is a scalar quantity and is normally written as simply the letter without the arrow. It is also sometimes written as the absolute value of the vector. (Magnitude of 𝑨 )= 𝑨 =𝐴

Displacement One of the simplest vectors is displacement, the change in position of an object. Consider the case of walking from the library to the ISA building. Your displacement is approximately 400 m to the northwest. The distance you walk depends on the path you took. Maybe you stopped at the Marshall center, or maybe you took the most direct route, annoying drivers as you cut diagonally across the roads.

Vector Addition Vector addition: 𝑪 = 𝑨 + 𝑩 Add vectors “head to tail.” The order doesn’t matter. 𝑪 = 𝑨 + 𝑩 = 𝑩 + 𝑨 When adding parallel vectors, the resulting magnitude is the sum of the two vector magnitudes. When adding antiparallel vectors, the resulting magnitude is the difference of the two vector magnitudes. When adding more than two vectors, they can be grouped in any combination and order.

Vector Subtraction Vector subtraction: 𝑫 = 𝑨 − 𝑩 Subtracting 𝑩 is the same as adding a negative 𝑩 . 𝑫 = 𝑨 + − 𝑩 The order does matter. 𝑫 = 𝑨 − 𝑩 ; 𝑬 = 𝑩 − 𝑨 𝑫 =− 𝑬

Multiplication of a Vector by a Scalar Multiplication by a scalar: 𝑮 =𝑐 𝑨 Resulting magnitude is the product of 𝑐 and 𝐴 Multiplying a unit vector 𝒙 by a scalar 𝐴 𝑥 gives a vector along the 𝑥 direction with magnitude 𝐴 𝑥 . This is the basis of the component form of vectors.

Vector Components If we set up an orthogonal coordinate system, we can express any vector in terms of its components along each of the axes. We can use trigonometry to find the magnitudes of the components. For a right triangle: sin 𝜃 = opposite hypotenuse cos 𝜃 = adjacent hypotenuse tan 𝜃 = opposite adjacent

Unit Vectors Unit vectors have a magnitude of one, and are distinguished by a caret or “hat.” In a coordinate system, unit vectors point along the positive axes. In a Cartesian coordinate system, there are three axes, 𝑥, 𝑦, and 𝑧. The corresponding unit vectors are 𝒙 , 𝒚 , and 𝒛 , or equivalently 𝒊 , 𝒋 , and 𝒌 . In component form, subscripts distinguish the component of the vector along each axis. 𝑨 = 𝐴 𝑥 𝒙 + 𝐴 𝑦 𝒚 + 𝐴 𝑧 𝒛 or, 𝑨 = 𝐴 𝑥 𝒊 + 𝐴 𝑦 𝒋 + 𝐴 𝑧 𝒌

Right-Handed Coordinate System Coordinate systems should be “right-handed.” Using your right hand, hold your thumb, index finger, and middle finger at right angles. Each digit will point in the positive direction: Thumb: 𝑥 axis Index finger: 𝑦 axis Middle finger: 𝑧 axis Note: There are other methods to determine the handedness of a coordinate system.

Vector Components Measuring the angle 𝜃 counterclockwise from the 𝑥 axis, we can find the components from the magnitude and direction. 𝐴 𝑥 =𝐴 cos 𝜃 𝐴 𝑦 =𝐴 sin 𝜃 Likewise, from the components, we can find the magnitude and direction. 𝐴= 𝐴 𝑥 2 + 𝐴 𝑦 2 𝜃= tan −1 𝐴 𝑦 𝐴 𝑥 There is always some ambiguity when using the inverse tangent function. Any two angles that differ by 180° will have the same tangent. By drawing the vector, it will be apparent which angle to use.

Vector Addition by Components Once vectors are expressed in terms of their components, addition and subtraction is trivial. The 𝑥, 𝑦, and 𝑧 components are each added/subtracted amongst themselves, without mixing. 𝑨 = 𝐴 𝑥 𝒙 + 𝐴 𝑦 𝒚 + 𝐴 𝑧 𝒛 𝑩 = 𝐵 𝑥 𝒙 + 𝐵 𝑦 𝒚 + 𝐵 𝑧 𝒛 𝑪 = 𝑨 + 𝑩 𝑪 =( 𝐴 𝑥 + 𝐵 𝑥 ) 𝒙 +( 𝐴 𝑦 + 𝐵 𝑦 ) 𝒚 +( 𝐴 𝑧 + 𝐵 𝑧 ) 𝒛 𝑫 = 𝑨 − 𝑩 𝑫 =( 𝐴 𝑥 − 𝐵 𝑥 ) 𝒙 +( 𝐴 𝑦 − 𝐵 𝑦 ) 𝒚 +( 𝐴 𝑧 − 𝐵 𝑧 ) 𝒛

Vector Addition Example

Scalar (Dot) Product Many physical relationships can be expressed by the product of vectors, but we cannot use ordinary multiplication with vectors. Multiplying vectors using the dot product results in a scalar quantity. The dot, or scalar product can be calculated using the magnitudes of the vectors and the angle between them. 𝑨 ∙ 𝑩 =𝐴𝐵 cos 𝜙 From this definition, it is clear that the dot product is at a maximum when 𝑨 and 𝑩 are parallel (𝜙=0°) and that the dot product is zero when 𝑨 and 𝑩 are perpendicular.

Scalar Product by Components To find the dot product by components, we look at the dot products of the unit vectors. Since the dot product of perpendicular vectors is zero, 𝒙 ∙ 𝒚 = 𝒚 ∙ 𝒛 = 𝒛 ∙ 𝒙 =0 But, the dot product of parallel vectors is the product of their magnitudes, 𝒙 ∙ 𝒙 = 𝒚 ∙ 𝒚 = 𝒛 ∙ 𝒛 =1 Multiplying this all out, 𝑨 ∙ 𝑩 =( 𝐴 𝑥 𝒙 + 𝐴 𝑦 𝒚 + 𝐴 𝑧 𝒛 )∙( 𝐵 𝑥 𝒙 + 𝐵 𝑦 𝒚 + 𝐵 𝑧 𝒛 ) 𝑨 ∙ 𝑩 = 𝐴 𝑥 𝐵 𝑥 + 𝐴 𝑦 𝐵 𝑦 + 𝐴 𝑧 𝐵 𝑧 We readily see that the dot product is commutative. 𝑨 ∙ 𝑩 = 𝑩 ∙ 𝑨

Meaning of Scalar (Dot) Product The dot product, 𝑨 ∙ 𝑩 , represents the product of the magnitude of 𝑨 and the projection of 𝑩 along the direction of 𝑨 . Likewise, it is also the product of the magnitude of 𝑩 and the projection of 𝑨 along the direction of 𝑩 .

Vector (Cross) Product Multiplying vectors using the cross product results in a vector. 𝑪 = 𝑨 × 𝑩 The resulting vector is perpendicular to both vectors 𝑨 and 𝑩 . The direction is given by the right hand rule. Point the index finger of your right hand along 𝑨 and curl your fingers towards 𝑩 . Your thumb is pointing in the direction of 𝑨 × 𝑩 . The magnitude of the cross product is. 𝑨 × 𝑩 =𝐴𝐵 sin 𝜙 The cross product is zero when 𝑨 and 𝑩 are parallel.

Vector Product by Components To find the cross product by components, we look at the cross products of the unit vectors. Since the cross product of parallel vectors is zero, 𝒙 × 𝒙 = 𝒚 × 𝒚 = 𝒛 × 𝒛 =0 But, using the right hand rule, 𝒙 × 𝒚 =− 𝒚 × 𝒙 = 𝒛 𝒚 × 𝒛 =− 𝒛 × 𝒚 = 𝒙 𝒛 × 𝒙 =− 𝒙 × 𝒛 = 𝒚 Multiplying this all out, and collecting terms: 𝑨 × 𝑩 =( 𝐴 𝑥 𝒙 + 𝐴 𝑦 𝒚 + 𝐴 𝑧 𝒛 )×( 𝐵 𝑥 𝒙 + 𝐵 𝑦 𝒚 + 𝐵 𝑧 𝒛 ) 𝑨 × 𝑩 = 𝐴 𝑦 𝐵 𝑧 − 𝐴 𝑧 𝐵 𝑦 𝒙 + 𝐴 𝑧 𝐵 𝑥 − 𝐴 𝑥 𝐵 𝑧 𝒚 +( 𝐴 𝑥 𝐵 𝑦 − 𝐴 𝑦 𝐵 𝑥 ) 𝒛 We readily see that the cross product is not commutative. 𝑨 × 𝑩 =− 𝑩 × 𝑨

Vector Product by Determinant We can also express the cross product in determinant form. 𝑨 × 𝑩 = 𝒙 𝒚 𝒛 𝐴 𝑥 𝐴 𝑦 𝐴 𝑧 𝐵 𝑥 𝐵 𝑦 𝐵 𝑧 Using cofactor (Laplace) expansion, we find the determinant. 𝑨 × 𝑩 =+ 𝒙 𝐴 𝑦 𝐴 𝑧 𝐵 𝑦 𝐵 𝑧 − 𝒚 𝐴 𝑥 𝐴 𝑧 𝐵 𝑥 𝐵 𝑧 + 𝒛 𝐴 𝑥 𝐴 𝑦 𝐵 𝑥 𝐵 𝑦 𝑨 × 𝑩 = 𝐴 𝑦 𝐵 𝑧 − 𝐴 𝑧 𝐵 𝑦 𝒙 − 𝐴 𝑥 𝐵 𝑧 − 𝐴 𝑧 𝐵 𝑥 𝒚 +( 𝐴 𝑥 𝐵 𝑦 − 𝐴 𝑦 𝐵 𝑥 ) 𝒛 𝑨 × 𝑩 = 𝐴 𝑦 𝐵 𝑧 − 𝐴 𝑧 𝐵 𝑦 𝒙 + 𝐴 𝑧 𝐵 𝑥 − 𝐴 𝑥 𝐵 𝑧 𝒚 +( 𝐴 𝑥 𝐵 𝑦 − 𝐴 𝑦 𝐵 𝑥 ) 𝒛

Meaning of Vector (Cross) Product The magnitude of the cross product, 𝑨 × 𝑩 , is the area of a parallelogram with sides 𝑨 and 𝑩 . From this, we clearly see that the area (and therefore the cross product) is zero when . 𝑨 and 𝑩 are parallel.

Products of Vectors Example

Chapter 1 Summary Units, Physical Quantities, and Vectors Idealized models – Know your assumptions Units SI units, prefixes, and unit consistency Uncertainty and significant figures Order of magnitude approximations – Is the answer reasonable? Vectors (magnitude and direction) and scalars (magnitude) Component notation and unit vectors Vector addition and subtraction (graphically and by components) Dot product: 𝑨 ∙ 𝑩 =𝐴𝐵 cos 𝜙 = 𝐴 𝑥 𝐵 𝑥 + 𝐴 𝑦 𝐵 𝑦 + 𝐴 𝑧 𝐵 𝑧 Cross product: 𝑨 × 𝑩 =𝐴𝐵 sin 𝜙 𝑨 × 𝑩 = 𝐴 𝑦 𝐵 𝑧 − 𝐴 𝑧 𝐵 𝑦 𝒙 + 𝐴 𝑧 𝐵 𝑥 − 𝐴 𝑥 𝐵 𝑧 𝒚 +( 𝐴 𝑥 𝐵 𝑦 − 𝐴 𝑦 𝐵 𝑥 ) 𝒛

Chapter 2 Outline Motion Along a Straight Line Velocity and Acceleration Average Instantaneous Graphical representation Motion with constant acceleration Kinematic equations Free fall Motion with varying acceleration Equations for position and velocity

Displacement in One Dimension First, we need to define a coordinate system. For one dimension, this just means choosing the origin and the positive direction. Displacement: ∆𝑥= 𝑥 2 − 𝑥 1 If the displacement is in the positive direction, ∆𝑥>0. If the displacement is in the negative direction, ∆𝑥<0.

Average Velocity in One Dimension The average velocity is the change in displacement divided by the time interval. 𝑣 av−x = ∆𝑥 ∆𝑡 = 𝑥 2 − 𝑥 1 𝑡 2 − 𝑡 1 Is this the same as the average speed? Keep in mind that with this definition, only the total displacement and the total time are taken into consideration.

Instantaneous Velocity in One Dimension The instantaneous velocity is the limit of the average velocity as the time interval approaches zero. 𝑣 𝑥 = lim ∆𝑡→0 ∆𝑥 ∆𝑡 = 𝑑𝑥 𝑑𝑡