Meeting times, textbook, homework, exam, grading policies
Written solutions, study groups, and plagarism.
Plan for what we'll cover this semester, and what will be learned.
Review mechanics from Physics 2170 or equivalent. For instance, in Halliday, Resnick, and Walker, look over Chapters 1 to 12.
In Taylor, read sections 1.1 and 1.2 for today, and 1.3 to 1.5 for Friday.
We will be studying classical mechanics in the Newtonian form this semester. We will discuss one dimensional motion with a variety of forces, projectile motion with resistive forces, harmonic motion, and orbits. Next semester's course, PHY5210, will delve into the Langrangian form of mechanics and a bit of special relativity and Hamiltonian mechanics.
Classical mechanics is one of the oldest branches of physics, yet still an active area of research today. Even when not directly the focus of research, an understanding of classical mechanics is essential to the design of many devices. For instance, atomic force microscopes rely on changes to the vibration of a macroscopic tip, governed by classical mechanics, to sense the atomic level forces in a sample, governed by quantum mechanics.
Along with mechanics, we will be learning mathematical techniques, dimensional analysis, approximation methods, and numerical techniques. Since these tools are applicable to many areas of physics, these topics are as important as the physics you will be learning.
Time, space (distance or location), and mass (weight) are quantities that we know from experience. We can sense the passage of time, we can observe distance, and we feel weight. Everyone has some understanding of these quantities before they begin the study of Physics.
Experience tells us that the motion of an object doesn't depend on our choice of how we measure the motion -- choice of starting point of measurement, unit of measurement, starting point in time, ... Nor does it depend on the direction of axes, when the motion occurs in more than one dimension. But we do know that displacement, velocity, and acceleration have both magnitude and direction. Mathematical objects with these same characteristics are vectors. When written in terms of vectors (vector notation), the rules of mechanics appear in a particularly simple form.
Vectors exist independently of a coordinate system, but for solving problems we normally need to express vectors in a particular coordinate system. In three dimensional space, a vector in a particular coordinate system is described by three numbers, the components of the system. For standard Cartesian coordinates, there are a number of equivalent notations for the three components of a vector: x-hat, y-hat, and z-hat; i-hat, j-hat, and k-hat; e1, e2, e3; and (x,y,z). We will use these notations, depending on convenience. In the future we will also see notations for coordinate systems other than Cartesian, namely polar/cylindrical, and spherical coordinates.
Here we review the rules of vector algebra. Now we have two types of objects, vectors and scalars (plain numbers). In these notes, vectors are represented by bold faced letters, like r, while scalars are denoted by normal, or italic letters, like a or r.
The following table contrasts the properties of scalars and vectors
| Operation | Scalars α and β | Vectors r and s |
| Addition (closure) | α + β exists | r + s exists |
| Commutative law | α + β = β + α | r + s = s + r |
| Associative law | α + (β + γ) = (α + β) + γ | r + (s + t) = (r + s) + t |
| Zero (0) exists | α + 0 = α | r + 0 = r |
| Negative exists | α + (-α) = 0 | r + (-r) = 0 |
| Multiplication (closure) | αβ exists | αr exists (mult. by scalar) |
| Commutative law | αβ = βα | αr = rα |
| Associative law | α(βγ) = (αβ)γ | α(βr) = (αβ)r |
| One (1) exists | α1 = α | 1r = r |
| Inverse | αα-1 = 1 | NA |
| Distributive law I | α(β + γ) = αβ + αγ | α(r + s) = αr + αs |
| Distributive law II | NA | (α + β)r = αr + βr |
| Scalar Product exists | NA | r⋅s = α |
| Commutative law | NA | r⋅s = s⋅r |
Vectors add with other vectors as scalars add with scalars; addition of a vector with a scalar is not defined. Notice that the inverse of a vector is not defined. This means that division by a vector is not defined!
The rules listed above are independent of representation of the vectors. Being practical, we will use vectors defined in (or that span) a real, 3-dimensional, Euclidean space, and these vectors will normally be represented as a vector in a 2 or3-dimensional coordinate system, either cartesian, polar, cylindrical, or spherical. In the cartesian representation, vectors are added and subtracted by adding or subtracting corresponding components.
"Multiplying" two vectors can be done in either of two ways, one that yields a scalar and one that yields another vector. In either case, the result is independent of the representation of the vectors. This is particularly important for the scalar product, since the result, being just a number, doesn't rely on the choice of representation of vectors. That is, the scalar product of any two vectors is a number, that depends on the vectors, but not on the coordinate system used to represent them.
The scalar (or inner or dot) product of two vectors yields a scalar, and is obtained by multiplying corresponding components (in any representation) and summing:
The length of a vector, represented by the vector written in normal text or surrounded by absolute value bars, is given by the square root of the scalar product with itself
Therefore, the final equality in the defintion of the scalar product is the usual result that the scalar product of two vectors equals the product of the lengths of the two vectors and the cosine of the angle between them.
The vector (or cross) product of two vectors yields another vector perpendicular to the original pair
with length
In Cartesian coordinates, the components of the resulting vector are:
Equivalently, this can be expressed as the determinant of a 3×3 matrix like
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i |
j |
k |
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t = det |
rx |
ry |
rz |
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sx |
sy |
sz |