Maths Help Please

TBirdUK · 21 Apr 2019 at 09:07

My brain is a bit slow today and needs some help.

I have a 10 inch diameter wheel which can give a speed of 4 mph at max revs - what would be the equation and result if I changed the diameter to 16 inches?

Thanks

h4rm0ny · 21 Apr 2019 at 09:22

The distance you travel is a factor of the circumference of the wheel (we're going to assume no slippage between the road and the tyre).

The formula for the circumference is Circumference = pi * diameter.

So the circumference of the 10" wheel is: 31.4"
And the circumference of the 16" one is: 50.2"

If the wheel rotates at the same speed in both cases (max revs is fixed), then your speed would increase by the same factor as the circumference has increased. I.e. (50.2/31.4) * 4mph.

However, we don't actually need to do all this. The moment that we see the formula for circumference is pi * diameter, we can see that the circumference is going to increase in direct proportion to the diameter. I.e. if the diameter increases by a factor of 1.6, then the circumference increases by a factor of 1.6. And indeed we can see that 50.2/31.4 is 1.6. So basically, just multiply your mph by the same factor the diameter has increased by.

4mph * (16/10) = 6.4mph.

If the formula for circumference were something like pi * diameter squared, then that wouldn't be so. It would be a square of the factor, and so on. Hope that helps.

lunar · 21 Apr 2019 at 09:28

I hope this is right.
At maximum revs the rotational motion is constant irrespective of the wheel size (ignoring the mass of the wheel). This motion is termed 'angular velocity, w', which defined as the rate at which an angle of the wheel is subtended. This value will be the same at max revs.
The equation that links linear velocity (v) and angular velocity (w) is:
V = w x r
Where r is the radius of the wheel. Half the diameter. I.e. 10 inch is 5 inch radius, so 16 inch is 8 inch radius.

Calculate w = v/r = 4/5 = 0.8

So, new speed, v = w x r = 0.8 x 8 = 6.4 mph

Dis86 · 21 Apr 2019 at 09:29

Depends. The speed may actually be lower than that calculated. Whatever is driving the wheel may have insufficient torque to do so.

lunar · 21 Apr 2019 at 09:34

Hence, the mass of the wheel becomes a factor. I think you them may need to use angular momentum and moment of inertia.

dl8860 · 21 Apr 2019 at 09:43

OP is this a purely theoretical mathematical question, or a practical real world problem?

lunar · 21 Apr 2019 at 09:45

Incidentally, the first responder mentioned velocity squared. This particularly applies to the drag force a vehicle experiences when travelling at higher speeds, like on motorways. At low speeds the drag force depends on velocity (not velocity squared).
On motorways, the optimum speed is about 60mph. Go faster than this will burn much more fuel as the 'velocity squared' factor kicks in.

TBirdUK · 21 Apr 2019 at 09:59

Cheers all, much appreciated - its for swapping out the 10 inch wheels of a mobility scooter for 16 inch ones when I take the drive and put it in my boys jeep - torque wont be an issue (I think) as the mobility scooted was designed to pull 20-22 Stone, him and the jeep will be about half that.
Thanks again

Orionaut · 21 Apr 2019 at 10:12

Amazed at how complicated people made this.

Pi has nothing to do with it!

Diameter is 1.6 times the original diamiter

Circumference is therefore 1.6 times that of the original circumference,

Therefore, for same RPM, speed is 1.6 times original speed!

IE 6.4

OTOH, I did score over 40 on the autism test in the post I made earlier. So I dont always realise when people are taking the **** or not! :/

Jonnybmac · 21 Apr 2019 at 10:33

This is where you define a mathematical model to assume certain things. you could go into all sorts really such as friction of bearings, surface and drag.

I would do it lunars way but H4rm0ny's way is interesting.

lunar · 21 Apr 2019 at 12:16

Orionaut said:
Amazed at how complicated people made this. Pi has nothing to do with it!

Diameter is 1.6 times the original diamiter

Circumference is therefore 1.6 times that of the original circumference,

Therefore, for same RPM, speed is 1.6 times original speed!

IE 6.4

OTOH, I did score over 40 on the autism test in the post I made earlier. So I dont always realise when people are taking the **** or not! :/

This is like the method used in gears.
My method was simply trying to explain the principle as the calculation was been performed.

h4rm0ny · 21 Apr 2019 at 12:44

TBirdUK said:
Cheers all, much appreciated - its for swapping out the 10 inch wheels of a mobility scooter for 16 inch ones when I take the drive and put it in my boys jeep - torque wont be an issue (I think) as the mobility scooted was designed to pull 20-22 Stone, him and the jeep will be about half that.
Thanks again

Ah, so it's a practical question. Then you should be a little wary of theoretical answers. Insert joke about spherical cows. Sounds like you have plenty of power behind it though so should be good.

Jonnybmac said:
I would do it lunars way but H4rm0ny's way is interesting.

My way is to multiply 4mph by 1.6!

Everything else is just explanation because I thought they were interested in how and why, rather than just whether their offspring will be accelerated away at ungodly speeds.

TBirdUK · 22 Apr 2019 at 00:13

luckily I have a variable speed throttle, so will adjust until he gets used to it - he had one, before I ripped all the gear out for another, that was about 6mph so all should be good - I am also going to install a remote 'kill' switch on the power, just in case!

Jonnybmac · 22 Apr 2019 at 00:27

h4rm0ny said:
Ah, so it's a practical question. Then you should be a little wary of theoretical answers. Insert joke about spherical cows. Sounds like you have plenty of power behind it though so should be good.

My way is to multiply 4mph by 1.6!

Everything else is just explanation because I thought they were interested in how and why, rather than just whether their offspring will be accelerated away at ungodly speeds.

Yeah, it is the most basic. It is easy to over complicate things by going with the equation you know is right rather than taking a step back and looking at the bigger picture of the question

Double07 · 22 Apr 2019 at 01:47

Orionaut said:
Amazed at how complicated people made this. Pi has nothing to do with it!

Diameter is 1.6 times the original diamiter

Circumference is therefore 1.6 times that of the original circumference,

Therefore, for same RPM, speed is 1.6 times original speed!

IE 6.4

OTOH, I did score over 40 on the autism test in the post I made earlier. So I dont always realise when people are taking the **** or not! :/

You assume someome can correlate the two. Not everyone knows that by increasing the diameter by 1.6 that curcumference also increases by the same factor.

LuckyBenski · 23 Apr 2019 at 18:28

TBirdUK said:
Cheers all, much appreciated - its for swapping out the 10 inch wheels of a mobility scooter for 16 inch ones when I take the drive and put it in my boys jeep - torque wont be an issue (I think) as the mobility scooted was designed to pull 20-22 Stone, him and the jeep will be about half that.
Thanks again

I would point out that torque DOES factor in - because at present your top speed may well be limited by, or at least partly involve, the load being driven. So you may find with half the load, top RPM is higher and top speed goes up more than you expect. Then again the bigger wheels would require more torque too, so YMMV. Literally

Sounds like you have it in hand with throttle adjustment and kill switches either way.