A certain hydraulic system is designed to exert a force 100 times as large as the one put into it. (a) What must be the ratio of the area of the slave cylinder to the area of the master cylinder? (b) What must be the ratio of their diameters? (c) By what factor is the distance through which the output force moves reduced relative to the distance through which the input force moves? Assume no losses to friction.
To find the answers to these questions, we need to apply the principles of Pascal's law, which states that the pressure applied to a fluid in a closed system is transmitted equally in all directions.
(a) What must be the ratio of the area of the slave cylinder to the area of the master cylinder?
We know that the hydraulic system is designed to exert a force 100 times as large as the one put into it. Since pressure is force divided by area, we can write the equation:
Slave pressure (P_slave) = Master pressure (P_master) * Force ratio
Now, using Pascal's law, we know that the pressure is transmitted equally in all directions. Therefore, we can write the equation:
P_slave = P_master
Considering that the area of the slave cylinder (A_slave) is different from the area of the master cylinder (A_master), we get:
Force ratio = A_slave / A_master
Therefore, the ratio of the area of the slave cylinder to the area of the master cylinder is equal to the force ratio. In this case, it is 100.
(b) What must be the ratio of their diameters?
The area of a circle is proportional to the square of its diameter. So, the area ratio is equal to the square of the diameter ratio. Let's assume D_slave and D_master as the diameters of the slave and master cylinders, respectively. The equation becomes:
(A_slave / A_master) = (D_slave^2 / D_master^2)
From part (a), we know that the area ratio is equal to the force ratio, which is 100. Therefore:
100 = (D_slave^2 / D_master^2)
Simplifying, we get:
D_slave^2 / D_master^2 = 100
Taking the square root of both sides, we find:
D_slave / D_master = 10
Therefore, the ratio of the diameters of the slave and master cylinders is 10.
(c) By what factor is the distance through which the output force moves reduced relative to the distance through which the input force moves?
The force exerted on the slave piston is directly proportional to the pressure and indirectly proportional to the area. Since the slave cylinder has a larger area, the force exerted on it will be greater than the force applied to the master cylinder. Therefore, considering that force equals pressure times area, and the pressure is the same in both cylinders, the greater force on the slave cylinder will result in a smaller distance the output force moves compared to the distance through which the input force moves.
Since the force is inversely proportional to distance, the factor by which the distance is reduced would be the ratio of the forces. From part (a), we know that the force ratio is 100. Therefore, the distance through which the output force moves would be reduced by a factor of 100 compared to the distance through which the input force moves.