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Algebra for Physics
The algebra that the physics chapters lean on. Rearranging a formula with its units, linear equations solved together, the quadratic equation and its two roots, ratios and scaling laws, inequalities and conditions, and the binomial approximation, each taught from a problem of the book.
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Why this chapter
Why this chapter exists
You may go on with the physics now. When a lesson needs this tool, it will tell you and bring you here.
Physics is written in formulas, and a formula helps only once it can be turned to give the quantity you want. The physics chapters use a small set of algebra again and again without stopping to teach it. This chapter teaches that set. It covers rearranging, equations solved together, the quadratic and its two roots, ratios and scaling laws, inequalities, and the binomial approximation. Every lesson starts from a problem of the book and ends by solving it.
You will learn
By the end of this chapter you will
- Rearrange any formula to make a chosen quantity the subject, carrying the units through, even when the unknown appears on both sides.
- Solve two or three linear equations together by elimination or substitution, and recognise equations that are not independent.
- Solve a quadratic by factors or by the formula, count its roots with \(b^2 - 4ac\), and keep only the roots that the physics allows.
- Use \(y_2/y_1 = (x_2/x_1)^n\) to answer "if \(r\) doubles, what happens?" for any power law.
- Write "at least", "at most" and "only if" as inequalities, and solve them, reversing the sign when multiplying by a negative number.
- Use \((1 + x)^n \approx 1 + nx\) for a small \(x\), and judge the size of its error.
Before you begin
Before you begin
You should be able to add, subtract, multiply and divide signed numbers and fractions, and use powers and roots such as \(x^2\) and \(\sqrt{x}\). The last lesson uses the derivative of a power once, from the chapter Differentiation. The rest needs no calculus.
Lessons
1
Rearranging a Formula
Make any quantity the subject of a formula by doing the same to both sides and undoing its operations in the reverse order. Keep the symbols until the last line, carry the units with the numbers, collect an unknown that sits on both sides, and use the identities that shorten the algebra.
Rearranging a formulaSymbols first and units through the algebraThe unknown on both sidesIdentities that shorten the algebra
2
Simultaneous Linear Equations
Two unknowns need two independent equations, three unknowns need three. Solve them by elimination or substitution, see each equation as a straight line, and use the method on the Atwood machine, a train of blocks, a collision and a network of cells.
Simultaneous linear equationsEach equation is a straight lineThree unknowns one at a time
3
The Quadratic Equation and Its Two Roots
Solve \(ax^2 + bx + c = 0\) by factors or by the formula, count the roots in advance with the discriminant \(b^2 - 4ac\), and read each root as a moment of a real motion, keeping the ones that make sense and rejecting a negative time.
The quadratic equation and its two rootsThe discriminant counts the rootsWhich root the physics keeps
4
Ratio, Proportion and Scaling
When \(y \propto x^n\), the constant cancels and \(y_2/y_1 = (x_2/x_1)^n\). Direct, inverse and inverse-square laws, "if \(r\) doubles", several factors at once, and the scaling laws of Kepler, Stokes, Stefan and Poiseuille.
Ratio proportion and scalingDirect inverse and inverse squareScaling laws of physics
5
Inequalities and Conditions
The phrases "at least", "at most" and "only if" written as inequalities, the rules for solving them, and the one step that reverses them. The conditions that forces impose, a string that only pulls, a floor that only pushes, friction that has a limit, and the band of values set by an absolute value.
Inequalities and conditionsConditions that forces imposeA band of allowed values
6
The Binomial Approximation
For a small \(x\) and any power \(n\), \((1 + x)^n \approx 1 + nx\). Where it comes from, how large its error is, how to bring an expression into its form, and its uses in physics, from \(g\) at a height and the expansion of areas to the kinetic energy of a slow body in relativity.
The binomial approximationHow large is the errorSmall changes by the binomial
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- absolute value
- The size of a number without its sign, written \(|x|\). \(|x| \le a\) means \(-a \le x \le a\).
- Atwood machine
- Two masses joined by a light string over a smooth, light, fixed pulley. Its two equations of motion are solved together for the acceleration and the tension.
- binomial approximation
- The rule \((1 + x)^n \approx 1 + nx\) for a small number \(x\) and any real power \(n\).
- constant of proportionality
- The fixed number \(k\) in \(y = kx^n\). It cancels whenever two cases are compared.
- discriminant
- The number \(b^2 - 4ac\) of a quadratic. It is positive for two real roots, zero for one, and negative for none.
- elimination
- Adding or subtracting multiples of equations so that one unknown cancels.
- identity
- An equation that is true for every value of its letters, such as \(a^2 - b^2 = (a - b)(a + b)\).
- independent equations
- Equations of which none can be made by adding or scaling the others. Only independent equations count towards fixing the unknowns.
- inequality
- A statement that one quantity is less than, or greater than, another, written with \(<\), \(\le\), \(>\) or \(\ge\).
- power law
- A law of the form \(y = kx^n\), in which one quantity is a fixed power of another.
- proportional
- Two quantities are proportional when their ratio is the same in every case, \(y = kx\). More generally \(y \propto x^n\) means \(y = kx^n\).
- quadratic equation
- An equation of the form \(ax^2 + bx + c = 0\) with \(a \ne 0\), in which the unknown appears squared and no higher.
- root
- A value of the unknown that satisfies an equation. A quadratic has at most two real roots.
- simultaneous equations
- Two or more equations that must all hold at once for the same values of the unknowns.
- subject
- The quantity that stands alone on one side of a formula. In \(l = gT^2/4\pi^2\), \(l\) is the subject.
- substitution
- Making one unknown the subject of one equation and putting that expression into another equation.
- weighted mean
- A mean in which each value counts in proportion to its weight. The final temperature of a mixture is the mean of the temperatures weighted by the heat capacities.