About Derek Bolton

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

Entries by Derek Bolton

Subtleties Overlooked in Friction Questions: Object Slides Down Ramp

December 6, 2022/0 Comments/in Physics Tutorials/by Derek Bolton

Problem statement (simplified) An object slides down a ramp at angle θ to encounter level ground. Both surfaces have kinetic friction: μ’ on the ramp, μ on the level. The object reaches the ground at speed u. What is its speed when first fully on the level? (Original is at https://www.physicsforums.com/threads/distance-a-block-slides-along-a-surface-with-friction-given-with-an-initial-velocity.1047556/.) There are several missing…

Corrections to MIT Open Courseware: Systems of Varying Mass

September 14, 2021/1 Comment/in Physics Tutorials/by Derek Bolton

Corrections to MIT Open Courseware 8-01sc classical mechanics, fall 2016 Applying Newton’s Laws to systems of Varying Mass PDF Course Link My concerns with the solutions provided on the cited MIT Open Courseware web page are, mostly: Despite arriving at the correct answers, the algebraic methods are invalid. As a result, they can be utterly…

Corrections to the Australian Physics Olympiad Questions and Solutions

September 1, 2020/3 Comments/in Physics Guides/by Derek Bolton

ref: asi.edu.au/programs/past-exams/physics-olympiad-past-exams/ Classification of issues: [1*] Quibble regarding the solution. [2*] Clearer problem description suggested. [3*] Alternative solution should be accepted [4*] Significant error, likely leading to incorrect marking. Tally matrix: Year 1* 2* 3* 4* 2019 3 0 0 0 2017 2 2 0 2 2016 1 1 2 0 2015 2 2 1…

Can Angles be Assigned a Dimension?

November 12, 2016/106 Comments/in Mathematics Articles/by Derek Bolton

1. Some Background on Dimensional Analysis … if you are not already familiar with it. 1.1 Dimensions Dimensional Analysis is a way of analyzing physics equations that consider 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…

Frequently Made Errors in Vectors – Elementary Use

March 4, 2016/2 Comments/in Physics Tutorials/by Derek Bolton

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…

Frequently Made Errors in Climate Science – The Greenhouse Effect

December 23, 2015/87 Comments/in Bio/Chem Articles/by Derek Bolton

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 that pass visible light through but block infrared can act as heat…

Frequently Made Errors in Heat: Elementary Level

June 1, 2015/3 Comments/in Physics Tutorials/by Derek Bolton

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…

Errors in Probability: Continuous and Discrete Distributions

May 30, 2015/1 Comment/in Mathematics Tutorials/by Derek Bolton

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…

Frequently Made Errors: Pseudo and Resultant Forces

May 26, 2015/12 Comments/in Physics Tutorials/by Derek Bolton

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 to correct them. 2. Applied versus Resultant The applied forces on an object are the specific forces exerted on it by…

Frequently Made Errors in Mechanics: Springs

May 23, 2015/3 Comments/in Physics Tutorials/by Derek Bolton

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…

Frequently Made Errors in Probability: Conditionals in Natural Language

May 22, 2015/2 Comments/in Mathematics Tutorials/by Derek Bolton

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:…

Frequently Made Errors in Mechanics: Momentum and Impacts

May 20, 2015/1 Comment/in Physics Tutorials/by Derek Bolton

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…

Frequently Made Errors in Mechanics: Hydrostatics

May 19, 2015/3 Comments/in Physics Tutorials/by Derek Bolton

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…

Some Conceptual Difficulties in the Roles of Variables and Constants

May 17, 2015/3 Comments/in Mathematics Tutorials/by Derek Bolton

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 the variables and…

Frequently Made Errors in Equation Handling

May 13, 2015/15 Comments/in Physics Tutorials/by Derek Bolton

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…

Frequently Made Errors in Mechanics: Kinematics

May 12, 2015/11 Comments/in Physics Tutorials/by Derek Bolton

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 are unnecessary. 1. Scalars and Vectors X “A scalar only has magnitude while a vector has magnitude and direction”…

Frequently Made Errors in Mechanics: Moments

May 10, 2015/7 Comments/in Physics Tutorials/by Derek Bolton

The term “moment” is used in various ways in Physics and Mathematics: Given a force and a reference point, the force has a moment (or torque) about the point. A rigid body has moments of mass of various orders (zeroth, first, second..) Similarly, moments of area, and moments of statistical distributions This page only…

Frequently Made Errors in Mechanics: Friction

May 8, 2015/8 Comments/in Physics Tutorials/by Derek Bolton

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…

Frequently Made Errors in Mechanics: Forces

May 6, 2015/35 Comments/in Physics Tutorials/by Derek Bolton

Notation: On 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 and…