A 100-kg person sits on a 5-kg bicycle.The total weight is borne equally by the two wheels of the bicycle.The tires are 2.0 cm wide and are inflated to a gauge pressure of 8.0 × 10⁵ Pa.What length of each tire is in contact with the ground? A)1.6 cm B)1.8 cm C)3.2 cm D)2.4 cm E)6.4 cm

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Quizplus · Accepted Answer

The total weight of the person and the bicycle is $100\, \text{kg} + 5\, \text{kg} = 105\, \text{kg}$. The force due to gravity acting on this total mass is $F = mg = 105\, \text{kg} \times 9.8\, \text{m/s}^2 = 1029\, \text{N}$. Since the weight is equally distributed between the two wheels, each wheel bears half the total force, so $F_{\text{per wheel}} = \frac{1029\, \text{N}}{2} = 514.5\, \text{N}$.The pressure $P$ is given by the force $F$ divided by the area $A$ in contact with the ground, $P = \frac{F}{A}$. The gauge pressure inside the tires is given as $8.0 \times 10^5\, \text{Pa}$, and this is the pressure exerted outward by the air inside the tire against the ground (and thus the pressure with which the tire pushes against the ground).Rearranging the pressure equation to solve for the area $A = \frac{F}{P}$, and knowing that the length of the tire in contact with the ground $L$ is related to the area by $A = L \times \text{width of the tire}$, we can solve for $L$.Given that the width of the tire is $2.0\, \text{cm} = 0.02\, \text{m}$, we have:$$L = \frac{F_{\text{per wheel}}}{P \times \text{width}} = \frac{514.5\, \text{N}}{8.0 \times 10^5\, \text{Pa} \times 0.02\, \text{m}} = \frac{514.5}{16000} = 0.032125\, \text{m} = 3.2\, \text{cm}$$Therefore, the length of each tire in contact with the ground is 3.2 cm.

A 100-Kg Person Sits on a 5-Kg Bicycle