Answer to the Question 09/96
The question was:A car participates in a race. Its tire explodes. What should be the the speed of the car, so that the tire does not become "flat"?
(7/98) The problem has been solved correctly by Itzhak Shapir (e-mail microwav@rafael.co.il). The answer is about 150 km/hour.
The solution:
Quarter of the weight of a car is supported by each wheel. The spped of the car must be large enough so the the centrifugal force on the part of the wheel which is in contact with the ground be equal to that part of the weight of the car. Here is the detailed solution by Shapir:
a. Assumptions on the car are that it has a mass M=1000kg, equally shared among wheels (i.e. 250kg per wheel)
b. Assumptions on the wheel are as follows:
(1) outer diameter is 0.5m;
(2) inner diameter is 0.3m;
(3) tire mass is 5kg, equally spread around its outer diameter;
(4) effective vertically oriented centripetal force is equally spread on
10cm of the tires footprint under its axes.
c. Method of calculation:
(1) A segment having mass m rotating around the axes with angular speed w
at a distance r from the center of movement will apply the force :
f=m*r*w**2o or f=m*v**2/r,
where v=w*r is the linear velocity. In case of
that tire v is the car's velocity.
(2) The mass of the effective tire footprint (10cm) is:m=5*0.1/(3.14*0.5) ~0.32kg
(3) In order to balance the 250kg weight (~2500N force) the equation will be:
2500 = 0.32*v**2/0.15 (0.15m is the flat position equals inner radius)
or v**2 = 2500*0.15/0.32 ~ 1250
so v = sqrt(1250) ~ 35 m/sec, which is about 120km/h
This is the speed to start recovering from flat tire.
d. Further calculation may be of the speed to maintain original tire inflation. Same assumptions on 10cm footprint and 250kg car mass on the wheel, but radius will be about 0.25m instead 0.15. In that case speed is calculated to be about 150km/h.
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