Fred & Barney's Car - Road Shape Analysis · RMPH 2013 #2

Physics · mechanics

Problem

Fred and Barney built a car having as "wheels" two identical square prisms. The wheels are perfectly suited to the road's shape (that is a periodical repetition of identical bumps), so that the center of mass of the car does not move vertically during the trip. The wheels never slip on the road. During the car's motion, the wheel's long edge always touches a "valley" of the road; road "heights" are reached by a line passing through the half of a rectangular face of the "wheels". In the figure is shown the motion of the lower side of the square cross-section of the wheel on a bump (the curve passing through the points

x_s

, T,

x_d

). The road is a periodic repetition of such a bump. The Ox axis of the coordinate system passes through valleys of the relief and the Oy axis passes through bump tip. AB is wheels side, T is the contact point between wheel and road and G is wheel axis position – in initial position. A'B',T' ,G' have the same meaning – but for a certain position. In cross-section the wheel axis is always on the same vertical line with the contact point between wheel and road. Task No. 2 - Road's shape 2.a. Give an explanation for this finding. 2.b. Determine the analytical form

y = y(x)

of cross-section of the road. *Hint: The derivative

y'(x)

of the function

y = y(x)

is the slope of the tangent to the graph of function in x. Consider known that

\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln x + \sqrt{x^2 - a^2} + C

. We recommend to use the notation

\text{ch}(x) = \frac{e^x + e^{-x}}{2}

;

\text{sh}(x) = \frac{e^x - e^{-x}}{2}

with property

\text{ch}^2(x) - \text{sh}^2(x) = 1

* 2.c. Determine the expression of the length (on horizontal) of a bump of road. 2.d. Determine the expression of minimum distance between wheel axles Fred's car. 2.e. Decide whether a machine wheeled regular hexagonal prism conveniently chosen could run on the road whose form was determined in 2.b., without vertical displacement of the center of mass of the wheel.

Browse the problem library