You spin me right round baby…

Newtonian Mechanics, Part One

Leopard, Going Postal
Newton, by William Blake; here, Newton is depicted critically as a “divine geometer”. This copy of the work is currently held by the Tate Collection
William Blake [Public domain]

Musings on objects and motion, part one, the basics

Recently a comment, that nobody read, was posted by one of our more trollish contributors trying to toll people about the world not be round or spinning, it went roughly thus:

“if the world is spinning how come we are not thrown off by centrifugal force? And why don’t we weight less at the equator than the poles?”

Now while the phrase obvious troll is obvious applies there is an important principle at stake here. Roughly stated, by myself as thus:

“if you don’t actually know the answer to a question, don’t dismiss the question or questioner as foolish”

Note: I’m not saying you have to answer the question, just that dismissing it out of hand when you yourself don’t actually know the answer puts you in the same group as the Thunderbergs of this world who denounce “climate change deniers” yet are unable to actually answer any of the points raised.

The result of this was a self-imposed challenge, to actually answer the question, but also to explain all the principals involved in a way that can be comprehended by anyone with what amounts to an understanding of GCSE mathematics and physics and has perhaps a smattering of interest in the world around them.

This is actually quite a deep subject the moment you start looking at not only what happens, but, how, why and how you can verify this yourself (because science should be repeatable and verifiable in order to be trusted) in reasonably basic ways.

That bloke who got hit by an apple

You may have heard of a certain English gentleman who is reputed to have been hit by a falling apple and had a bit of a revelation by the name of Isaac Newton (1642 – 1726). In 1686 he published the seminal “Philosophiæ Naturalis Principia Mathematica” which wackipedia tells me translates as “Mathematical Principles of Natural Philosophy”. This work and the principles within govern much of what we now know about how and why things move the way things do.

That bloke who looks like he stuck his fingers in an electrical socket

You may also have heard of a certain German gentleman who was an all-round damned clever chappy by the name of Albert Einstein (1879 – 1955). He took what Newton did, along with others, and supercharged it by taking it to the extremes and trying to break it, then coming up with new theories etc. to cover those situations.

While Einstein’s work is both amazing and important, in every day terms we do not move things that fast generally or work with things that small or that large so with a slight coughing sound we will not be covering it here, just the basic principles established by Newton after some yob threw an apple at him.

Part 1 – the basics

In order to look at how a spinning world works we first need to consider how the “laws” established by Newton work, and before we can do that we need to look at what they are.

I’m going to use the definitions provided by wackypedia as these are slightly easier to relate to than the ones Newton himself wrote (in Latin, told you he was clever).

Three laws of motion

First Law

In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity. Unless acted upon by a force.

Second Law

In an inertial frame of reference, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma (it is assumed here the mass is constant)

Third Law

When a body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.


In the above there are a few terms, in italic, that are worth expanding upon before we try to go further.

Inertial frame of reference: this is establishing a ground rule, e.g. when you say something is moving at a velocity of “x”, what is this velocity relative to? For example when you consider your cars speed when driving you are doing so relative to the ground, the fact the ground is also moving relative to the sun doesn’t matter.

Velocity: velocity is a vector, it describes the motion of an object in terms of both its magnitude (speed) and direction (heading), so if you know the velocity you know how fast you are moving and in what direction.

Force: this is nothing to do with a galaxy a long long time ago etc. this is a way of describing any interaction that would, if unopposed, result in a change in the motion of an object. forces can push or pull. E.g. when you pull on a door handle you are applying a force to the door handle which you are probably hoping will open the door. Forces are also vectors, and represented under the SI system of measurement with the symbol F and measured in newtons.

Mass: mass is a physical property of “things”. It is a measure of the resistance to acceleration and determines the gravitational attraction the thing exerts upon other things. Note this is related to, but not the same as an objects weight. Mass is a scalar quantity, under the SI measurement system denoted with the symbol m. the basic unit of measurement is the kilogram (kg).

Body: basically a way of describing a “thing”, a body will have a mass and a velocity and is a thing upon which forces can act. While a body can be any shape or size it is usually represented as a point object to which other properties can be applied, this point being the centre of mass of the object

Vector: a vector is something that has both a direction and a magnitude, typically represented in equations in bold face text. Mathematical functions can be applied to vectors, e.g. addition and subtraction, multiplication etc, see the average Monday morning for a far better description of how this can be done than I can manage. Examples of vectors we will use are velocity and forces

Scalar: a scalar is something that has only a single dimension often just magnitude, e.g. mass, temperature, speed, pressure. Typically represented in equations using italic face text.

Definitions 2, units of measurement

As noted above there are a few units of measurement we need to consider when looking at objects and objects in motion, these are worth defining just for clarity – these can easily be looked up and should not be contentious for anyone demanding “proof” as they are really just a way or describing properties.

Mass: a fundamental property (that is to say not depending upon anything else), represents the resistance to acceleration (also known as inertia) as noted previously, measured in kilograms (scalar). Usually represented as: m.

Time: a fundamental property, represents the passage of time, how long things take etc, measured in seconds (scalar). Usually represented as: t.

Distance/Displacement: a fundamental property, represents the size of something, e.g. length, width, measurement between two points, measured in metres (scalar). Usually represented as: s

Speed: a derived property, measuring the rate of change of distance, measured in metres per second (m/s or ms-1), quite literally how many metres of distance are covered in a second (scaler)

Velocity: Speed, but with both magnitude (speed) and direction (specified relative to the frame of reference), also measured in metres per second. Not as confusing as it could be as when considering motion we are talking about the vector velocity not the scalar speed unless indicated otherwise. Usually represented as u (initial) or v (final)

Acceleration: the rate of change of velocity (or speed), this measure as the name suggests how quickly the velocity of a body is changing – this could be a change in magnitude or direction, or both. Measured in metres per second, per second (m/s/s or ms-2). As an example an object accelerating at 1ms-2 in a stated direction will increase is velocity in that direction by 1ms-1 every second. Note acceleration can also be negative, a useful way of stating the object is slowing down, or the direction component can be reversed. Usually represented as a or where the acceleration is due to the force of gravity g.

Maffs! (Constant Acceleration)

There are some basic equations (known as the SUVAT equations) that are worth knowing, these can all be confirmed by simple experimentation and measurement for anyone who is interested, again nothing contentious here, this is just about working out how things move. Note all of these assume constant acceleration, where acceleration varies you are going into calculus and I’m avoiding that as its not a Monday, where you have simple changes (e.g. acceleration a1 for t1 and then acceleration a2 for t2 you can resolve this in two stages)

v = u + at

this is useful to determine a final velocity (v) knowing the initial velocity (u), the acceleration (a) and the elapsed time (t).

Example, an object moving at 1 ms-1 initially, accelerates at 1ms-2 for five seconds, its final velocity will be 6 ms-1.

s = ut + ½at2

this one is useful for determining how far something will move (its displacement, s) in a period of time (t) given its initial velocity (u) and its acceleration (a). note if the object isn’t moving the first part resolves to zero, if its not accelerating the second part resolves to zero.

Example, an object moving at 1 ms-1 initially, accelerates at 1ms-2 for five seconds, its displacement will be 17.5m from its starting location. Note the first equation also informs us of its final velocity

s = ½(u + v)t

this one is useful for determing how far something will move (its displacement, s) in a period of time (t) when you know the initial and final velocity and the time period.

Example, and object is initially moving at 1ms-1 and eventually at 6ms-1 for 5 seconds, its displacement will be 17.5 from its starting location.

v2 = u2 + 2as

this one is useful to determine final velocity when you know the starting velocity, acceleration and distance covered.

Example, an object initially moving at 1ms-1 covers 17.5 metres while accelerating at 1ms-2, its final velocity is: 6ms-1

s = vt – ½at2

this one is useful when you know the final velocity, time and acceleration but not the initial velocity and need the displacement.

Example, an objects final velocity is 6ms2 it has been accelerating at 1ms-2 for 5 seconds, it has covered 17.5m

The above are all pretty much basic exam questions related to the physics of motion, these should stir dim memories in anyone who studied GCSE physics or above, no idea if “combined science” covers it though.

This covers constant linear acceleration (i.e. acceleration in the straight line), there are similar equations covering constant rotational acceleration (i.e. changing the speed of an objects rotation)

Now, suitably armed with this we are ready to proceed to the next part, where we will start to look at Newton’s laws and what they actually mean

© Leopard 2020

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