Principles of physics 1

Course notes for PHYS-1408: Principles of Physics 1. Texas Tech University, Whitacre College of Engineering. Spring 2023.

Additional resources:

• College Physics 2e on OpenStax

1 Physics and Measurement

1.1 Standards of Length, Mass, and Time

Fundamental qty.	SI unit
length	meter
mass	kilogram
time	second

We can identify length as the distance between two points in space. In October 1983 the meter was redefined as the distance traveled by light in vacuum during a time interval of $\frac{1}{299,792,458}$ second.

The mass of an object is related to the amount of material that is present in the object, or to how much that object resists changes in its motion. The kilogram is defined as the mass of a specific platinum–iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sèvres, France.

Before 1967, the standard of time was defined in terms of the mean solar day. One second is now defined as $9,192,631,770$ times the period of vibration of radiation from the cesium-133 atom.

Length, mass, and time are examples of fundamental quantities. Most other variables are derived quantities, those that can be expressed as a mathematical combination of fundamental quantities. Common examples are:

Area (a product of two lengths):

Speed (a ratio of distance to a time interval):

Density (a ratio of mass to volume):

Prefixes for powers of ten

Power	Prefix	Abbreviation
$10^{-24}$	yocto	y
$10^{-21}$	zepto	z
$10^{-18}$	atto	a
$10^{-15}$	femto	f
$10^{-12}$	pico	p
$10^{-9}$	nano	n
$10^{-6}$	micro	$\mu$
$10^{-3}$	milli	m
$10^{-2}$	centi	c
$10^{-1}$	deci	d
$10^{3}$	kilo	k
$10^{6}$	mega	M
$10^{9}$	giga	M
$10^{12}$	tera	T
$10^{15}$	peta	P
$10^{18}$	exa	E
$10^{21}$	zetta	Z
$10^{24}$	yotta	Y

1.2 Modeling and Alternative Representations

A model is a simplified substitute for the real problem that allows us to solve the problem in a relatively simple way.

• Geometric model: a geometric construction that represents the real situation and analyze
• Simplification model: details that are not significant in determining the outcome of the problem are ignored
• Analysis model: general types of problems that we have solved before
• Structural model: used to understand the behavior of a system that is far different in scale from our familiar experience

A representation is a method of viewing or presenting the information related to the problem.

• Mental representation
• Pictorial representation
• Simplified pictorial representation
• Graphical representation
• Tabular representation
• Mathematical representation

1.3 Dimensional Analysis

The method of dimensional analysis is very powerful in solving physics problems. Dimensions can be treated as algebraic quantities. For example, quantities can be added or subtracted only if they have the same dimensions. Furthermore, the terms on both sides of an equation must have the same dimensions. By following these simple rules, you can use dimensional analysis to determine whether an expression has the correct form.

Quantity	Area ($A$)	Volume ($V$)	Speed ($v$)	Acelleration ($a$)
Dimensions	$L^2$	$L^3$	$\frac{L}{T}$	$\frac{L}{T^2}$
SI units	$\text{m}^2$	$\text{m}^3$	$\frac{\text{m}}{\text{s}}$	$\frac{\text{m}}{\text{s}^2}$

We can use brackets $[\enspace]$ to denote the dimensions of a physical quantity, e.g. $[v] = \frac{L}{T^2}$.

1.4 Conversion of Units

Like dimensions, units can be treated as algebraic quantities that can cancel each other. The remaining unit is our desired result.

1.5 Estimates and Order-of-Magnitude Calculations

An order of magnitude is a power of $10$ determined as follows:

1. Express the number in scientific notation, with the multiplier of the power of $10$ between $1$ and $10$ and a unit.
2. If the multiplier is less than $\sqrt{10}$ (approx. $3.16$), the order of magnitude of the number is the power of $10$ in the scientific notation. If the multiplier is greater than $\sqrt{10}$, the order of magnitude is one larger than the power of $10$ in the scientific notation.

We use the symbol $\sim$ for “is on the order of.”

Number	Expression	Order of Magnitude
0.0086	$8.6 \times 10^{-2}$	$-2$
0.0021	$2.1 \times 10^{-3}$	$-3$
31	$3.1 \times 10^{1}$	$1$
32	$3.2 \times 10^{2}$	$2$

1.6 Significant Figures

The number of significant figures in a measurement can be used to express something about the experimental uncertainty.

Rules to identify significant figures in a number:

• Non-zero digits within the given measurement or reporting resolution are significant
• Zeros between two significant non-zero digits are significant (significant trapped zeros)
• Zeros to the left of the first non-zero digit (leading zeros) are not significant
• Zeros to the right of the last non-zero digit (trailing zeros) in a number with the decimal point are significant, if they are within the measurement or reporting resolution
• Trailing zeros in an integer may or may not be significant, depending on the measurement or reporting resolution
• An exact number has an infinite number of significant figures
• A mathematical or physical constant has significant figures to its known digits

Addition & subtraction

When numbers are added or subtracted, the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum or difference: e.g. $123 + 1.1 = 124$ (the first term has the smallest number of decimal places).

Multiplication & division

When multiplying several quantities, the number of significant figures in the final answer is the same as the number of significant figures in the quantity having the smallest number of significant figures. The same rule applies to division: e.g. $0.0032 \times 356.3 = 1.1$ (the first term has only 2 significant figures).

Experimental uncertainty

A propagation of uncertainty allows us to estimate the uncertainty in a result from the uncertainties in the measurements used to calculate that result. Uncertainty in a measured quantity is quantitative statement of precision. Measurements of a quantity gives rise to two values: main value $V$ and uncertain amount $\delta V$.

We report uncertainty in a measurement in one of two equivalent ways:

• Absolute uncertainty: uncertainty in a measurement expressed using the relevant units (fractional uncertaity) $\frac{\text{Absolute diff}}{\text{True value}}$
• Relative uncertainty: uncertainty of a measurement compared to the size of the measurement (percent uncertainty) $\frac{\text{Absolute diff}}{\text{True value}} \times 100 \%$

Calculating uncertainty:

• When measurements with uncertainties are multiplied or divided, add the relative uncertainties to obtain the percent uncertainty in the result.
• When measurements with uncertainties are added or subtracted, add the absolute uncertainties to obtain the absolut uncertainty in the result.
• If a measurement is taken to a power, the percent uncertainty is multiplied by that power to obtain the percent uncertainty in the result.

Cf.:

• Physics Bootcamp: 1.4 Uncertainty, Precision, Accuracy
• Physics Bootcamp: 1.5 Propagation of Uncertainty

2 Motion in One Dimention

2.1 Position, Velocity, and Speed of a Particle

The displacement of a particle is defined as its change in position in some time interval. As the particle moves from an initial position $x_i$ to a final position $x_f$, its displacement is given by

It is very important to recognize the difference between displacement and distance traveled. Distance is the length of a path followed by a particle.

The average velocity of a particle is defined as the particle’s displacement $\Delta x$ divided by the time interval $\Delta t$ during which that displacement occurs.

In everyday usage, the terms speed and velocity are interchangeable. In physics, however, there is a clear distinction between these two quantities.

The average speed of a particle, a scalar quantity, is defined as the total distance $d$ traveled divided by the total time interval required to travel that distance

2.8 Freely Falling Objects

The average velocity between two instants is given by

3 Vectors

Cf. Vectors.

3.1 Coordinate Systems

A Cartesian coordinate system is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length: $(x, y)$

A polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction: $(r, \theta)$

To convert polar coordinates to Cartesian coordinates:

• $x = r\cos(\theta)$
• $y = r\sin(\theta)$

To convert Cartesian coordinates to polar coordinate:

• $r = \sqrt{x^2 + y^2}$
• $\theta = \arctan(\frac{y}{x})$

3.2 Vector and Scalar Quantities

Cf. Vectors: Vector and scalar quantities.

3.2 Basic Vector Arithmetic

Cf. Vectors: Vector operations.

3.4 Components of a Vector and Unit Vectors

Cf. Vectors: Unit vectors.

Directional cosines

The directional cosines of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction.

Orthonormal basis $= \{ \hat{i}, \hat{j}, \hat{k} \}$ because $\hat{i} \perp \hat{j}$, $\hat{j} \perp \hat{k}$, and $\hat{k} \perp \hat{i}$.

For vector

$\vec{R} = r_x \hat{i} + r_y \hat{j}, + r_z \hat{k}$

whose magnitude is expressed as

$R = \lVert \vec{R} \rVert = \sqrt{r_x^2 + r_y^2 + r_x^2}$

the directional cosines are

• $r_x = R \cos \alpha = \cos \alpha = \frac{r_x}{R} \therefore \alpha = \cos^{-1}{\frac{r_x}{R}}$
• $r_y = R \cos \beta = \cos \beta = \frac{r_y}{R} \therefore \beta = \cos^{-1}{\frac{r_y}{R}}$
• $r_z = R \cos \gamma = \cos \gamma = \frac{r_z}{R} \therefore \gamma = \cos^{-1}{\frac{r_z}{R}}$

and it follows that

$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$

$\vec{R} = (R, \alpha, \beta, \gamma)$

• R has 3 independent degrees of freedom
• The angles are subjected to one constraint