20.33 Using this blog to study mechanics

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I have written this post to help students who are studying mechanics. My blog is recommended as a source of information to first-year undergraduate students studying mechanics at a UK university. Unfortunately, they have found it difficult to relate the structure of the blog to the list of topics in a conventional mechanics course. So I have written this post for them and other students studying mechanics at school, college or university.

I haven’t included vibrations, waves, fluid mechanics or quantum mechanics in this post because they’re not usually included in mechanics courses. If you’ve heard of Lagrange’s equations of motion or Hamiltonians, they’re not included for the same reason and because I haven’t written any posts about them.

If posts about using my blog to study other topics would be useful, let me know by posting a comment.

In the sections below, I often refer to a simple explanation of an idea before giving a more detailed (often mathematical) explanation. I suggest that you always read the simple explanation first, for the reasons given in post 17.25.

Mass, length and time

The concepts of mass, length and time are fundamental to mechanics. They are discussed and the units used to measure them are described in posts 16.12 and 16.13.

Displacement is related to the concept of length; it is a vector this means that it has direction as well as magnitude. More details are given in the first two paragraphs of post 17.4.

Velocity and acceleration

Velocity and acceleration are defined using the concepts of displacement and time. They are introduced simply, without using the mathematical concepts of differentiation or vector algebra in post 16.12 and the first three paragraphs of post 16.13.

To find out about differentiation and vector algebra see “Mathematical methods” at the end of this post.

When you understand differentiation and vector algebra, you will be able to understand velocity and acceleration in more detail by reading post 17.4.

Significant figures in calculations

You will now be able to calculate velocity and acceleration but must be careful to give your answers to a sensible number of significant figures as described in post 16.7.

Units in equations and dimensional analysis

The use of units to check equations is described in post 17.41.

The use of dimensional analysis to derive equations is described in post 17.42; the example given involves fluid mechanics but it isn’t difficult to understand (I hope!) and links are given to information on the concepts involved.

Relative velocity

Movement must be measured relative to an observer, as explained in post 16.4. A more difficult example is given in post 16.9.

Force (Newton’s laws of motion)

Force and its properties are defined by Newton’s laws of motion that are introduced, with no mathematics, in post 16.2. More details are given in posts 16.4, 16.12 and 16.13, in which the only mathematics used is simple algebra. The concept of force involves the concept of momentum that explained in post 16.13.

More mathematical definitions are given in the tables (“Translation” column) in post 17.39. For information about the mathematics involved see “Mathematical methods” at the end of this post.

Newton’s third law of motion is used to explain rocket propulsion in post 17.26.

The effect of the direction in which a force acts is described in post 16.50.

Gravity

Newton’s law of gravitation is described, with no vector algebra, in post 16.16. Gravity causes an object with mass to have a weight, as described in post 16.17. But mechanics would be easier if the word “weight” didn’t exist and we said “the force exerted by gravity”.

More details on gravitational fields, using some vector notation are given in post 17.24.

Falling, as a result of gravity, is explained in post 17.20; this explanation involves the mathematical concept of integration (see “Mathematical methods” below).

Centre of gravity is described in posts 17.21, 17.22 and 17.23. Post 17.21 involves vector algebra ; post 17.23 also involves the mathematical concept of integration (see “Mathematical methods” below).

The speed required to escape the earth’s gravitational field is discussed in post 17.27.

The role of gravity in the motion of satellites is described in post 17.28.

Centre of gravity (centre of mass)

Centre of gravity is described in posts 17.21, 17.22 and 17.23. Post 17.21 involves vector algebra; post 17.23 also involves the mathematical concept of integration (see “Mathematical methods” below).

Friction

The laws of friction are explained in post 16.19.

The involvement of friction in sliding is described in post 17.35.

Work, energy and power

Simple explanations of these concepts are given in posts 16.20, 16.21 and 16.23.

A more detailed explanation of work, involving integration and vector algebra (see “Mathematical methods” below) is given in post 17.36.

The energy of a stretched spring is described in post 16.49.

Collisions (elastic and inelastic)

Collisions are described in post 17.30.

Compression, tension and springs

The behaviour of springs is described in post 16.49 and used to explain the concepts of tension and compression.

Motion in a circle (rotational motion)

The concepts of angle, angular speed and angular acceleration are described in posts 17.11 and 17.12. This description uses the mathematical concept of differentiation (see “Mathematical methods” below).

When observing rotational motion, we need to be aware of the effects of aliasing (post 16.14).

Centripetal and centrifugal forces

Centripetal force is explained in post 17.13. This description uses the mathematical concept of differentiation and two-dimensional vector algebra (see “Mathematical methods” below).

The role of centripetal force in the motion of satellites is described in post 17.28.

The role of centripetal force in turning corners is described in post 17.34.

Centrifugal force is described in post 17.29. Centrifugal force is more difficult to understand and some people who write about it (on the web and, sometimes, in books) appear to misunderstand it. It may help you to read posts 16.4 and 16.9 first.

Torque

A simple explanation of torque is given in post 17.10. A student once complained to me that this way of explaining torque wasn’t fair because he shouldn’t be expected to be able to use a spanner (a “wrench” in American English). I think that you should be able to use a spanner if you’re studying mechanics!

A more detailed explanation of torque, involving vector algebra (see “Mathematical methods” below) is given in post 17.37.

An example of using the concept of torque is given in post 17.22.

Rotational inertia (moment of inertia)

This topic is explained in post 17.38.

Comparison between translational and rotational motion

The relationship between the concepts involved in translational motion and rotational motion is described in post 17.39. This topic is not covered in most books but it can help in solving problems involving rotational motion.

Rolling

Rolling is described in post 17.40. It is important to realise that a rolling object has more kinetic energy than the same object when it is moving with the same speed but not rolling.

Statics

When an object is not moving, in the frame of reference of an observer, we say that it is in equilibrium. This must mean that the vector sum of the torques applied to the system must be zero (post 17.22) and that the vector sum of all the forces applied to the system must be zero. So, if you understand dynamics (the motion of objects) statics is simple.

Mechanics of Materials

Stress, strain and Young’s modulus are described in post 20.2.

Hooke’s law is explained in post 20.3.

Poisson’s ratio is explained in post 20.5. You may then find that some of the information given in your course textbook is incorrect.

Elasticity is explained in post 20.6. Note that you shouldn’t confuse elasticity with compliance.

Also you shouldn’t confuse stiffness and strength – see post 20.9.

Toughness is described in post 20.10 – it isn’t the same as strength.

The elasticity of rubber is described in post 20.11. This involves the concept of entropy – links are given for further information on this, and other, topics in this post.

Fracture is described in post 20.13.

Shear is described in post 20.19.

Isometric compression and bulk modulus are discussed in post 20.20.

The relationships between Young’s modulus, shear modulus, bulk modulus and Poisson’s ratio are derived in posts 20.21 and 20.22.

Mathematical methods

To find out about vectors read post 16.50; this uses the concept of force to explain the idea of a vector because this concept makes some of the properties of vectors easier to understand. Then read posts 17.2 and 17.3.

The dot (scalar) product of vectors is explained in appendix 2 of post 17.13.

The cross (vector) product of vectors is explained in post 17.37.

To find out about differentiation read post 17.4.

To find out about integration read post 17.19.

 

I thank Celia Hukins and Daniel Espino for independently explaining to me the difficulty of finding a topic in my blog. This is the first stage in trying to rectify the problem. The next will be an index.

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