# Motion of a point on a tyre

Gold Member

## Homework Statement

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?

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Simon Bridge
Homework Helper
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".

• 1 person
Gold Member
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
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

• 1 person
Gold Member
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
Mentor
Cycloid.

Pretty easy to describe with a parametric equation.

• 1 person
haruspex
Homework Helper
Gold Member
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?

• 1 person
BvU
Homework Helper
2019 Award
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.

• 1 person
Simon Bridge
Homework Helper
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)?

• 1 person
Gold Member

BvU
Homework Helper
2019 Award
Does that mean you're OK ? Then I can provide the link ! • 1 person
Gold Member
It seems that point P is travelling faster than v, (≈4/3) is it so?

BvU
You mean it eventually gets ahead of the wheel ? 