Motion of a Point on a Tyre: Finding the Curve and Formulas

[bobie]

Homework Statement

Could you tell me how to find an article that deals with this motion:
a car (wheel =r) is traveling at speed v , a point P on the tyre describes what curve ? is it harmonic motion or what? what are its formulas?

 

[Simon Bridge]

You can figure it out for yourself though ... get something round, like a jar lid, put a dot on it and roll it along a ruler ... see? If you put the dot on the edge, then you can mark out the actual shape on a bit of paper.

Anyway - what you want is called a "locus".

 

[bobie]

You can figure it out for yourself though ...

Thanks,
It's a semi-elliptical curve with a= π (r) and b = 2 (r).
But how do I find the equation of motion of P? average speed is v , but at points -a and a it is =0,
what speed at point b?

 

[Chestermiller]

If the tire is traveling at speed v, what is the angular velocity of the tire about its axis? If the tire were just spinning but not traveling forward, do you know how to work out the x and y velocity components at any point during the rotation? For the car moving forward, the motion of the point is the same as the rotational movement plus a forward movement.

Chet

 

[bobie]

do you know how to work out the x and y velocity components at any point during the rotation?

I do not, any link? Thanks

 

[Borek]

Cycloid.

Pretty easy to describe with a parametric equation.

 

[haruspex]

I do not, any link? Thanks

First consider a wheel rotating about its centre at rate ω. Take that centre as origin in polar coordinates. Consider a point on it which is at radius = a, theta = 0 at time t=0. Where is it, in polar coordinates, at time t? What's that in Cartesian?
Now add in the fact that the centre is moving in the x direction at speed v = ωr. What does that do to the position of the point?

 

[BvU]

A link would be a spoiler. Write down x(t) and y(t) for circular motion around a fixed point. Now let the fixed center move in the x direction in such a way that it proceeds $2\pi R$ sideways per revolution.

Haha, five helpers jumping in! I pass.

 

[Simon Bridge]

I'm with BvU on this - divide the motion of the point into x and y.
Have the wheel moving in the x direction with speed v.

If v=0, what are the equations x(t) and y(t)?

 

[bobie]

Thanks for your help.

 

[BvU]

Does that mean you're OK ? Then I can provide the link !

 

[bobie]

It seems that point P is traveling faster than v, (≈4/3) is it so?

 

[BvU]

You mean it eventually gets ahead of the wheel ?

I see I made a small error when posting the cycloid link. Sorry. Now to business:

As you can see it moves sideways $2\pi R$ per revolution of the wheel, so no getting ahead, fortunately. And the average horizontal speed is v.
Instantaneous speed follows from differentiation. Good exercise ! Your 4/3 turns out to be even higher!