6.15) You wish to design a hydraulic system so that a force put into it would be increased by a factor of 100.
a. What must be the ratio of the area of the slave cylinder to the area of the master cylinder?
b. What must the ratio of thr diameter of the slave cylinder to the diameter of the master cylinder be?
c. By what factor is motion reduced in the system?
Note: Pls provide me an answer with solution/formula. Thanks!
I am about to scream. Please try do do this yourself before asking us.
To determine the answers to these questions, we can use Pascal's law, which states that the pressure applied to a fluid in a closed system is transmitted equally in all directions. In a hydraulic system, this means that when a force is applied to the master cylinder, it is transmitted to the slave cylinder and amplifies the force.
a. The pressure in the system is equal to the force divided by the area. So, to increase the force by a factor of 100, the numerator of the provided formula should be multiplied by 100. This means that the ratio of the area of the slave cylinder (As) to the area of the master cylinder (Am) must be 100:1. Mathematically, this can be expressed as:
As / Am = 100 / 1
b. The area of a cylinder is given by the formula: A = π * r^2, where A is the area and r is the radius of the cylinder. Since the radius and diameter of a cylinder are related by the formula: d = 2r, we can determine the ratio of the diameters of the slave cylinder (Ds) to the master cylinder (Dm) using the ratio of their areas. Mathematically, this can be expressed as:
(π * Ds^2) / (π * Dm^2) = As / Am = 100 / 1
By simplifying, we get:
(Ds / Dm)^2 = 100 / 1
Taking the square root of both sides, we obtain:
Ds / Dm = 10 / 1
Therefore, the ratio of the diameters of the slave cylinder to the master cylinder must be 10:1.
c. The motion in a hydraulic system is reduced by the inverse ratio of the areas of the cylinders. In this case, the area ratio is 100:1, so the motion is reduced by a factor of 1/100. Therefore, the motion is reduced by a factor of 0.01.
In summary:
a. As / Am = 100 / 1
b. Ds / Dm = 10 / 1
c. The motion is reduced by a factor of 0.01.