Final answer:
The diameter of a hydraulic piston cylinder is calculated using the formula d = sqrt[(4×Area) / π], and in a hydraulic system with a force ratio of 100:1, the ratio of the wheel cylinder to the pedal cylinder areas is 100:1, the diameter ratio is 10:1, and the distance moved by the output force is reduced by a factor of 100.
So option (D) is the correct answer.
Step-by-step explanation:
To find the diameter of a hydraulic piston-cylinder when provided with the area, we can use the formula d = sqrt[(4×Area) / π]. This formula derives from the area of a circle, A = πr², where r is the radius of the circle. To solve for the diameter (d), which is twice the radius (d = 2r), we rearrange the formula for area to solve for the diameter. The correct steps would involve dividing the area of the piston by π and then taking the square root, which is then multiplied by 4 to obtain the diameter. For example, if the calculated area of the piston is 25 cm², the diameter would be calculated as sqrt[(4× 25 cm²) / π], which equals approximately 5.64 cm.
In the context of a hydraulic system that exerts a force 100 times as large as the input force, the ratio of areas is directly related to the force exerted due to the principle of Pascal's law, which states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of its container. Hence, if a hydraulic system is designed to exert a force 100 times as large as the input force, the ratio of the area of the wheel cylinder to the area of the pedal cylinder must be 100:1. The ratio of their diameters, following the relationship between area and diameter (Area = π(d/2)²), would be the square root of the area ratio, so the ratio of their diameters would be 10:1. Finally, due to the conservation of energy and the fact that pressure is maintained, the distance through which the output force moves is reduced by the same factor as the area ratio, so by a factor of 100.