Overview

News

Lectures

Problems

Lecture Summaries

Click on each heading to download a pdf version of the lecture slides.

Click on these links to scroll down to lecture 1, lecture 2 or lecture 3.

Lecture 1: Basic Strategy

Approaches to problem-solving:

Recipe-based: follow a pre-defined series of steps which should eventually lead to a solution

• works well for a specific class of problem
• cannot be generalised to other problems, even if quite closely related
• not, in general, transferable to real-world situations
• Conclusion: use where appropriate, but not universally applicable

Equation-based: try to remember an equation that seems to include the right variables, and apply it

• if applied without first understanding the physics, absolutely no guarantee that the equation is actually relevant!
• as different quantities can be represented by same symbol, equation may not even really refer to same variables (e.g. m in astronomy may be either mass or magnitude)
• Conclusion: thoroughly bad idea, use only if utterly desperate (and even then, apply checks to ensure that at least your quantities have the right dimensions and behave in sensible ways)

Systematic: work out what's going on, identify applicable physical principles, then invoke relevant physical laws to solve problem

• requires thought, so likely to be slower than the recipe-based approach if the problem is one for which you have a recipe
• general – can be applied to any type of problem, even one you've never seen before
• transferable – can be applied to problems outside physics
• does require practice, and understanding of physical principles
• Conclusion: the most difficult and probably least intuitive approach, but the one most likely to work for the majority of problems

The systematic approach to problem-solving

1. Model the situation, using diagrams to help you think through exactly what is happening and what physical laws are involved.
2. Formulate the problem, converting your conceptual model into equations using the appropriate laws of physics.
3. Solve the equations, first symbolically, and then inserting numbers if required.
4. Check that the result makes sense: does it have the correct dimensions, does it behave well in easy-to-analyse special cases, is it numerically reasonable?

Key ingredients:

• Draw a diagram – this will (almost) always help you to see what is going on
• Work in symbols – this allows you to check the dimensions of your equation, and to see what happens if you use different numbers (checking special values)
• Always remember to check your result – everybody makes mistakes, but smart people find and fix them before they do any harm!
• Practise – it's the only way to improve your skills.

Lecture 2: Dimensional Analysis and Special Cases

Dimensional Analysis

Principle: if you add, subtract or equate variables, they must have the same dimensions (or units)

Uses:

• checking algebra – if you have derived an expression for, say, a force, it had better have the dimensions of force
• checking memory – if you are not sure you have remembered an equation correctly, checking dimensions may help (but see below)
• deducing functional dependence – if you know (or can guess) what variables your desired quantity is likely to depend on, constructing a combination with the right dimensions will tell you the form of the dependence
• making order-of-magnitude estimates – if you can deduce the dependence of your quantity on fundamental constants such as c, e, G etc., you can estimate its numerical value by assuming that the constant of proportionality is 1

Limitations:

• insensitive to pure numbers: πr² and 4πr² have the same dimensions
• getting the dimensions right won't stop you from getting the units wrong: hc/λ has the correct dimensions for energy (check!), but if you have λ in nm instead of m you will still get the wrong answer

Method:

• Express all quantities in terms of basic dimensions mass, length, time, charge (or current) and temperature
• Collect powers
  • if checking a result, confirm that the powers match
  • if deriving functional dependence, solve resulting set of simultaneous equations
• Recall that arguments of functions are pure numbers
  • so if your equation contains sin(kx) or e^kx, dimensions of k must be 1/length

Dimension checks are always worth doing: never take very long, catch many common algebraic mistakes (e.g. forgetting to square something)

Special Values

Many problems are easy to understand in special cases

• maybe when one quantity is very large or very small (extremes)
• maybe when two are equal (symmetry)

Also it is usually easy to see the effect of changing a variable

• will increasing x increase or decrease y?

Use both of these to check whether your result is acting in a sensible way

• can find errors that dimensional analysis can't find, e.g. upside-down ratios, missing minus signs

Scaling

Use ratios to go from one numerical result to another.

• you don't need to know the value of any numerical constant
• complementary to dimensional analysis (which gives you functional dependence but not numeric constants)
• can also be used to simplify calculations (and hence reduce risk of making mistakes)

Lecture 3: Expansions and Estimation

Power series expansions of common functions can be useful in simplifying calculations

Applications:

• derivations may rely on them, e.g. small angle approximation for simple pendulum
• can also make numerical calculations simpler to do and easier to check
• in some cases may even be required to achieve necessary precision (because a very large number may cancel out)

Know the most useful ones:

• sin θ = θ - θ³/3! + ...
• cos θ = 1 - θ²/2! + ...
• (1 + x)ⁿ = 1 + nx + ...

In general

• f(x+Δx) = f(x) + Δx f'(x) + (Δx)² f''(x)/2! +(Δx)³ f'''(x)/3! + ....
  where f'(x) = df/dx etc.

Other useful checks for numerical calculations:

Recognise when the final value is "obviously wrong"

• speeds > c
• scale wrong for context of question (e.g. interatomic distances should not come out in centimetres – and neither should interstellar distances!)
• not in accord with experience (if calculation refers to everyday events) – e.g. braking distance for a car really shouldn't come out as three miles (though it might for a supertanker)

Count up powers of 10

• e.g. suppose you're asked to estimate how much solar energy falls on 1 m² of the Earth's surface (perhaps for a question on solar cells)
• you look up the Sun's luminosity, 3.8x10²⁶ W, and the Earth-Sun distance, 1.5x10¹¹ m, and apply equation flux = L/4πr²
• so: the 3.8 and 4 approximately cancel, 10²⁶/(10¹¹)² = 10^26-22 = 10⁴, and the πx1.5x1.5 must be nearly another factor of 10
• conclude result should be about 1000 W m^-2
  (it's actually 1370 or so)

This is a good way to spot forgotten powers (e.g. forgot to square r in the above example)

Don't assume that a number is right "because that's what the calculator says" – errors in entering calculations into calculators are a common source of lost exam marks!