Motion of a point on a tyre

1. Apr 9, 2014

bobie

1. The problem statement, all variables and given/known data
Could you tell me how to find an article that deals with this motion:
a car (wheel =r) is travelling at speed v , a point P on the tyre describes what curve ? is it harmonic motion or what? what are its formulas?

2. Apr 9, 2014

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".

3. Apr 9, 2014

bobie

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?

4. Apr 9, 2014

Staff: Mentor

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

5. Apr 9, 2014

bobie

I do not, any link? Thanks

6. Apr 9, 2014

Staff: Mentor

Cycloid.

Pretty easy to describe with a parametric equation.

7. Apr 9, 2014

haruspex

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?

8. Apr 9, 2014

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.

9. Apr 9, 2014

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)?

10. Apr 10, 2014

bobie

11. Apr 10, 2014

BvU

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

12. Apr 11, 2014

bobie

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

13. Apr 11, 2014

BvU

You mean it eventually gets ahead of the wheel ?

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.