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towards James Clerk Maxwell 4: a detour into dimensional analysis and Newton’s laws

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nutshell Newton

Canto: Getting back to J C Maxwell, I’m trying to learn some basic physics, which may or may not be relevant to electromagnetism, but which may help me to get in the zone, so to speak.

Jacinta: Yes, we’re both trying to brush up on physics terms and calculations. For example, acceleration is change in velocity over time, which is hard to put in notation form in a blog post, but I can steal it from elsewhere

{\displaystyle {\bar {\mathbf {a} }}={\frac {\Delta \mathbf {v} }{\Delta t}}.}

in which the triangle represents ‘change in’. Now velocity is a vector quantity, therefore so is acceleration – it’s a particular magnitude in a particular direction. So imagine a car that goes from stationary to, say 50 kms/hour in 5 seconds, what’s the acceleration? According to the formula, it’s 50 – 0 kph/5 seconds, or 10kph/sec, which we can write out as a change of velocity of ten kilometres per hour per second.

Canto: So every second, the velocity of the car is increasing by 10 kilometres per hour. I’m trying to picture that. It’s quite hard.

Jacinta: Okay while you’re doing that, let’s introduce dimensional analysis, so that we reduce everything to the same dimension, sort of. I mean, we have hours and seconds here, so let’s take it all to seconds. I won’t be able to do this properly without an equation-writing plug-in, which I can’t work out how to get without paying. Anyhow..

10 kms/hour.second.1/3600 hour/second. Cancelling out the hours, you get 10 kms/3600 seconds squared, or 1/360 km/s2

Canto: I wonder if there’s a way of hand-writing equations in the blog, that’d be more fun and easy. So can you briefly explain dimensional analysis?

Jacinta: Well physical quantities are often measured in different units – for example, quantities of time – time is called the base quantity – are measured in seconds, hours, days etc. So, it’s just a matter of getting such measurements to be commensurate, so that an equation can be simplified – all in seconds, or all in metres, when they can be. Though actually it’s more complicated than that, and I’ve probably got it wrong.

Canto: So talking of brushing up on stuff, or actually knowing about stuff for the first time, I thought it might be good to go back to Newton, his three laws of motion, in written and mathematical form.

Jacinta: Go ahead.

Canto: Well, the first law, which really comes from Galileo, is often called the law of inertia. Newton formulated it this way, in the Principia (translated from Latin):

Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.

And as Sal Khan and others point out, Newton is talking about an unbalanced force, one that isn’t matched by an equal and opposite force (which would be a balanced force – see Newton’s third law). This law doesn’t come with a mathematical formula.

The second law, which I filched from The Physics Classroom, can be stated thus:

The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.

It’s famous formula is this:

Fnet = m • a

It can be written different ways, for example simply F =m.a, or with the vector sign (an arrow) above force (F) and acceleration (a), showing the same direction, but it’s certainly important to explain net force here. It’s essentially the sum of all the forces acting on the mass, in vector or directional terms. It’s this net force that produces the acceleration.

So to the third law, and this is how Newton presented it, again translated from Latin:

To every action there is always an equal and opposite reaction: or the forces of two bodies are always equal and are directed in opposite directions.

It’s often stated in this ‘wise proverb’ sort of way: ‘for every action there’s an equal and opposite reaction’.

Jacinta: What goes around comes around.

Canto: That’s more of a wise-guy thing. Anyway, the best formula for the third law is:

FA = −FB

where force A is the action and force B the reaction. This law is sort of counter-intuitive and also sort of obvious at the same time! I think it’s the most brilliant law. Sal Khan gives a nice extra-terrestrial example of how you might utilise it. Imagine you’re in outer space and you’ve been cut off from your spaceship and are accelerating away from it. To save yourself, take something massive, if you can, something on your suit or a tool you’re carrying, and push it hard away from you in the opposite direction to the ship, and this should send you accelerating back to the ship. But make sure your aim is true!

Jacinta: Okay, so this seems to have taken us absolutely no closer to Maxwell’s equations.

Canto: Well, yes and no. It makes us think of forces and energy, albeit of a different kind, and it makes us think in a logical, semi-mathematical way. but we’ve certainly got a long way to go…

References

https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion

Khan Academy physics

https://www.physicsclassroom.com/class/newtlaws/Lesson-1/Newton-s-First-Law

https://www.livescience.com/46558-laws-of-motion.html

https://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law

Written by stewart henderson

June 17, 2019 at 9:52 pm