suppose that you are designing an unpowered race car for the annual downhill der

Question

suppose that you are designing an unpowered race car for the annual downhill derby. you have a choice of wheels, such as thin disc wheels or solid spherical wheels.

mgh=(1/2)mv2+(1/2)I2

1) show that     v2= 2gh/1+I*                       WHERE I* =I/mR2,

and thus  I* is (and hence the smaller the wheel's moment of inertia) the faster the wheel will roll down the hill. R is radius. take g = 32ft/s2 and assume that the vertical height of the incline is h=100ft

2)show that ,whatever the wheel's design, the maximum velocity a circular wheel can attain on this incline is 80ft/s

3)which wheel will make the car go fastest downhill race car?

4) what is your conclusion?

Explanation / Answer

1) mgh=(1/2)mv2+(1/2)I2 ----> 1

Moment of inertia of four Wheels = I (car has 4 wheels)

putting I* =[I/mr2],
V2 +I*xr22= 2gh from eqn 1

putting = V/r (r term cancels out)

V2 +I*xV2 = 2gh

V2(1+I*) = 2gh

V2 =  2gh/(1+I*)

2) if is at its minimum I* then the maximum can be found

maximum traslational velocity of a circular wheel =  [2gh/(1+I*)]

condition I* > 0 because it should have mass around its axis

least possible tending value is 0

so Vmax =  (2gh) = 80 ft/s

3)  For V to be maximum -->I* should be minimum for a wheel

Solid spherical wheel has I* = 4* 2/5 = 1.6

thin disc wheels  I* = 4* 1/2 = 2

from given choice of wheels its mimimum for Solid spherical wheel

4) max soeed reached by this race car= [2gh/(1+I*min)] --->which is possible from given choice

Vmax = 49.614 ft/s

So reducing the moment of inertia of wheels makes the car faster

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