Taylor 1.3 - 1.4 (today) and 1.5-1.6 (Monday)
We were discussing vector operations: addition, subtraction, multiplication by a scalar.
| 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. The scalar (or dot) product of two vectors yields a scalar, and is obtained by multiplying corresponding components and summing:
where we use the non-bold name with no subscript to represent the length of the vector, r=|r|, and
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
| | | i | j | k | | | |
| t = det | | | rx | ry | rz | | |
| | | sx | sy | sz | | |
Most of the laws of physics are relations of the change in a quantity over time or space, and how one quantity changes with respect to another is expressed as a derivative. Since we've decided that the laws of physics will use vectors, we must be able to differentiate vectors. Differentiation of vectors is defined in the same manner as for scalars:
where
For our purposes, we will assume the limit exists and is well behaved.
Differentiation of vectors has the usuaul distributive properties of scalar derivatives,
and obeys the chain rule for the derivative of a scalar times a vector, the scalar product, and the vector product,
and
When the vector is expressed in Cartesian coordinates (where the axes are fixed and don't change with time), then the derivative of the vector is obtained by differentiating each of its components. That is, for
where x, y, and z can be functions of time, then its time derivative is given by
The derivative of a vector expressed in other coordinate systems (polar, cylindrical, or spherical) is more complicated. Because the coordinate axes are not fixed, they can change with time, and their derivatives must be included as well.
A shorthand notation for the time derivative is commonly used in mechanics:
that is, the quantity (vector or scalar) with a dot above. A second time derivative is denoted with two dots:
For this course we will use the "classical" or Newtonian view of time: time flows uniformly everywhere, and all observers agree on the elapsed time between two events. Different observers may have their clocks zeroed differently. For most problems on Earth, this definition is adequate. However, this picture of time is incorrect, and we see how it is fixed when we discuss special relativity next semester.
Two types of reference frames: inertial and non-inertial. In inertial reference frames, Newton's first law holds. We generally want to use inertial reference frames, and will devote a chapter to non-inertial reference frames later in the semester. For most problems outside of this chapter, we can regard the Earth's surface as an inertial reference frame.
Mass represents an objects tendency to resist motion, its "inertia". The text discusses an interesting problem in defining mass, namely the difference between "graviational" mass and "inertial" mass. The most commonly used method of determining mass, a balance, actually determines "gravitational" mass, how strongly an object is gravitationally attracted to the Earth. Experiments have confirmed the equivalence of gravitational and inertial mass to about a part in a trillion, and, for this course, we will assume they are absolutely equivalent.
We will normally be using SI units. The fundamental units for mechanics are:
| quantity | unit |
|---|---|
| length | meter (m) |
| time | second (sec, s) |
| mass | kilogram (kg) |
From introductory physics you should have developed a picture of what force is. Some forces are easy to recognize, you can feel if you push on a wall and feel the wall push back; others are more subtle, like the normal forces and frictional forces encountered in problems with inclined planes.
The text discusses some subtle issues with measuring forces. You are encouraged to read this. I will assume that we can measure forces in the unit of the newton, 1N = 1 kg⋅m/s².
A body in motion tends to remain in motion, a body at rest tends to remain at rest.
In the absence of forces, a particle moves with constant velocity v.
For any particle of mass m, the net force F on the particle is always equal to the mass m times the particle's acceleration:
The net force F acting on a particle is always equal to the time rate of change of the momentum p = mv:
The last two forms will be used in these notes to indicate time derivatives.
Note that the first law is really a special case of the second law: in the absence of forces, the net force is zero, and for a particle of constant mass the second law says that m(dv/dt) = 0 or that the velocity is constant.
A differential equation is simply an expression that relates derivatives to something. You've already encountered the simplest differential equations, and they are solved by simply integrating. As an example, consider the case of a particle acted on by a constant force in the x direction.
As written, Newton's laws are true in inertial frames only. Inertial frames are those in which the first law is true. Generally we will solve problems only in inertial frames, but in a later chapter we will explore how to solve problems in non-inertial frames. This is useful to solve certain problems, like the effects of the Earth's rotation on a pendulum or projectile.
The first two laws are valid in a large range of problems, as long as energy values are large relative to planck's constant (macroscopic systems), speeds are significantly less than the speed of light, and gravity is modest. These other cases are the domains of quantum mechanics, and special and general relativity. This still leaves a large range of problems to be addressed by classical mechanics.
© 2007 Robert Harr