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HairyButtock

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I'm trying to derive an expression for the normal forces on a cars tyres traveling around a circular banked track. Initially I was going to assume the centre of gravity was located longitudinally and laterally centre between all four tyres (so front and rear had the same forces on them). Also the car will be traveling at constant angular velocity.

However even with an unbanked track, I seem to be missing something, I know that the normal force must be velocity dependent (since it is going to decrease on the inside tyres and increase on the outside tyres as velocity increases).

In the image uploaded the axis of rotation is to the left.

Taking moments about the centre of gravity you get...

[tex] \sqrt{\frac{x^{2}}{4}+h^{2}}·(N_2sinθ-N_1sinθ-F_1cosθ-F_2sinθ)=0 [/tex]

and I know F

as well as the coefficient of friction being the same for both tyres.

I'm struggling to combine these two to give a velocity dependent expression for N

Confusingly when I take moments about the base of each tyre I get the same value for N

[tex] \sqrt{\frac{x^{2}}{4}+h^{2}}·\frac{mgsin\phi }{x} [/tex]

which is not velocity dependent or correct.

Cheers for any help.

EDIT-Sorry, I can't figure out how to move this post over to the homework/coursework-type questions forum or how to delete it.

However even with an unbanked track, I seem to be missing something, I know that the normal force must be velocity dependent (since it is going to decrease on the inside tyres and increase on the outside tyres as velocity increases).

In the image uploaded the axis of rotation is to the left.

Taking moments about the centre of gravity you get...

[tex] \sqrt{\frac{x^{2}}{4}+h^{2}}·(N_2sinθ-N_1sinθ-F_1cosθ-F_2sinθ)=0 [/tex]

and I know F

_{1}+F_{2}=mrω^{2}as well as the coefficient of friction being the same for both tyres.

I'm struggling to combine these two to give a velocity dependent expression for N

_{1}& N_{2}.Confusingly when I take moments about the base of each tyre I get the same value for N

_{1}& N_{2}after rearranging...[tex] \sqrt{\frac{x^{2}}{4}+h^{2}}·\frac{mgsin\phi }{x} [/tex]

which is not velocity dependent or correct.

Cheers for any help.

EDIT-Sorry, I can't figure out how to move this post over to the homework/coursework-type questions forum or how to delete it.

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