Entries by Derek Bolton

Can Angles be Assigned a Dimension?

1. Some Background on Dimensional Analysis … if you are not already familiar with it. 1.1 Dimensions Dimensional Analysis is a way of analysing physics equations that considers only the qualitative dimensions – mass, length, time, charge .. – of the quantities involved, not the values that they take in the given problem.  This is not to be […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Vectors – Elementary Use

  A vector has magnitude and direction. Pictorially, a vector can be imagined as a location in n-dimensional space relative to some fixed origin.  The magnitude is represented by the distance from the origin to the point. In Mechanics and Introductory Physics we mostly deal with 2- and 3-dimensional vectors. Some physical entities that behave […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Climate Science – The Greenhouse Effect

1.What is meant by “The Greenhouse Effect”? Many gases, such as H2O, CO2, CH4, are transparent to visible light but absorb and emit parts of the infrared spectrum.  Most of the visible light reaching the Earth’s surface gets re-emitted, eventually, as infrared.  Media which pass visible light through but block infrared can act as heat […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Heat – Elementary Level

1. Heat, Work, Internal Energy and Kinetic Energy “If heat is the motion of molecules, why isn’t it Kinetic Energy?” In everyday use, we may think of a hot body as containing heat, but in Physics Heat and Work refer to the transfer of energy from one body to another. Correspondingly, Internal Energy and Kinetic […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Probability – Continuous and Discrete Distributions

1. Classifying as discrete, continuous or mixed These statements (or equivalents) can be found in authoritative-seeming websites: X “A random variable is either discrete or continuous.” X “A random variable is continuous if it can take infinitely many values.” A random variable may be discrete, continuous or mixed.  The same adjective is applied to its […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors – Pseudo and Resultant Forces

  1. Real versus Fictitious Pseudo, or “fictitious“, forces can arise when a non-inertial frame of reference is used. Using a non-inertial frame makes the usual force/acceleration laws fail. Pseudo forces must be added in to correct them. 2. Applied versus Resultant The applied forces on an object are the specific forces exerted on it […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Mechanics – Springs

  1. Springs in Series “A spring of constant ##k_1## is connected in series with a spring of constant ##k_2##.  What is the spring constant for the composite spring?” Attempted answer: When the tension in the first spring is T, the tension in the second spring will also be T.  The two extensions will be […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Probability – Conditionals in Natural Language

  1. Turning a verbal condition into Algebra An actual thread…. “A study of auto accidents has found that 40% of all fatal accidents are attributed to texting drivers…” Attempt 1: X P(F|T) = .4 No, that says if an accident occurs while texting there is a 40% chance of its being fatal. Attempt 2: […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Mechanics – Momentum and Impacts

    An impact is an impulse (change of momentum) that involves arbitrarily large forces acting very briefly. These result in near-instantaneous changes in speed. In general, work will not be conserved. 1. Examples of impacts Most textbook impacts are collisions, but anything involving a sudden change in speed is an impact. For example, a […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Mechanics – Hydrostatics

  1. Archimedes’ Principle X “When a body is placed in a liquid, the weight of the body equals the weight of the liquid displaced” That will only be true if the body is floating.  Here’s what Archimedes wrote: ✓ “Any object, wholly or partially immersed in a fluid, is buoyed up by a force […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Conceptual Difficulties in the Roles of Variables

1. Variables and Constants When is a constant not a constant?  When it varies. In the standard equation ##y=ax+b##, we are used to thinking of x and y as variables, a and b as constants.  But it is just an algebraic relationship between four unknowns.  What is it that makes two of them variables and […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Equation Handling

  1. Algebra versus Arithmetic When numerical values are provided as inputs in a question, it is tempting to plug these into the equations straight away.  Resist!  Create variable names for these constants and work wholly symbolically until the final step.  There are many advantages: If you make a mistake, it takes less time to […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Mechanics – Kinematics

    Kinematics is the subset of dynamics that only concerns itself with time, displacement, velocity and acceleration. A problem is a kinematics problem if it involves those quantities, but consideration of masses, forces, and work is unnecessary. 1. Scalars and Vectors X  “A scalar only has magnitude while a vector has magnitude and direction” […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Mechanics – Moments

    Given a force and a reference point, the force has a moment (or torque) about the point. 1. Magnitude of Torque X “The magnitude of the moment of a force applied at a point P about a point Q is the magnitude of the force multiplied by the distance from P to Q” […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Mechanics – Friction

  1. Direction of the normal Definition: The normal force that body A exerts on body B is that force of minimum magnitude which suffices to prevent penetration of its surface. X “the normal is at right angles to the horizontal” This mistake tends to arise when one of the bodies presents a sharp corner […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge

Frequently Made Errors in Mechanics: Forces

    Notation: In this page, a circumflex signifies an average. 1. Forces as vectors A force is a vector, i.e. has magnitude and direction. But it also has a point of application, which vectors in general do not have.  A force can be thought of as a vector pair, one vector specifying its magnitude […]

Masters in Mathematics. Interests: climate change & renewable energy; travel; cycling, bushwalking; mathematical puzzles and paradoxes, Azed crosswords, bridge