Answer:
The additional weight and mass needed for lifting the other piston slowly is 2500 N and 254.92 kg, respectively.
Explanation:
By means of the Pascal's Principle, the hydraulic lift can be modelled by the following two equations:
Hydraulic Lift - Before change
![P = \frac{F}{A}](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7BF%7D%7BA%7D)
Hydraulic Lift - After change
![P + \Delta P = \frac{F + \Delta F}{A}](https://tex.z-dn.net/?f=P%20%2B%20%5CDelta%20P%20%3D%20%5Cfrac%7BF%20%2B%20%5CDelta%20F%7D%7BA%7D)
Where:
- Hydrostatic pressure, measured in pascals.
- Change in hydrostatic pressure, measured in pascals.
- Cross sectional area of the hydraulic lift, measured in square meters.
- Hydrostatic force, measured in newtons.
- Change in hydrostatic force, measured in newtons.
The additional weight is obtained after some algebraic handling and the replacing of all inputs:
![\frac{F}{A} + \Delta P = \frac{F}{A} + \frac{\Delta F}{A}](https://tex.z-dn.net/?f=%5Cfrac%7BF%7D%7BA%7D%20%2B%20%5CDelta%20P%20%3D%20%5Cfrac%7BF%7D%7BA%7D%20%2B%20%5Cfrac%7B%5CDelta%20F%7D%7BA%7D)
![\Delta P = \frac{\Delta F}{A}](https://tex.z-dn.net/?f=%5CDelta%20P%20%3D%20%5Cfrac%7B%5CDelta%20F%7D%7BA%7D)
![\Delta F = A\cdot \Delta P](https://tex.z-dn.net/?f=%5CDelta%20F%20%3D%20A%5Ccdot%20%5CDelta%20P)
Given that
and
, the additional weight is:
![\Delta F = (25\,m^{2})\cdot (100\,Pa)](https://tex.z-dn.net/?f=%5CDelta%20F%20%3D%20%2825%5C%2Cm%5E%7B2%7D%29%5Ccdot%20%28100%5C%2CPa%29)
![\Delta F = 2500\,N](https://tex.z-dn.net/?f=%5CDelta%20F%20%3D%202500%5C%2CN)
The additional mass needed for the additional weight is:
![\Delta m = \frac{\Delta F}{g}](https://tex.z-dn.net/?f=%5CDelta%20m%20%3D%20%5Cfrac%7B%5CDelta%20F%7D%7Bg%7D)
Where:
- Additional weight, measured in newtons.
- Additional mass, measured in kilograms.
- Gravitational constant, measured in meters per square second.
If
and
, then:
![\Delta m = \frac{2500\,N}{9.807\,\frac{m}{s^{2}} }](https://tex.z-dn.net/?f=%5CDelta%20m%20%3D%20%5Cfrac%7B2500%5C%2CN%7D%7B9.807%5C%2C%5Cfrac%7Bm%7D%7Bs%5E%7B2%7D%7D%20%7D)
![\Delta m = 254.92\,kg](https://tex.z-dn.net/?f=%5CDelta%20m%20%3D%20254.92%5C%2Ckg)
The additional weight and mass needed for lifting the other piston slowly is 2500 N and 254.92 kg, respectively.