In the hydraulic system shown in the figure, the piston on the left has a diameter of 4.5cm and a mass is 3.0kg.
If the density of the fluid is 710kg/m^3 , what is the height difference h between the two pistons in m?
I checked the other similar questions like this and tried to solve the question using the
Pressure (P) = pgh, but still could not get the right answer.
To solve this question, we need to use the principles of Pascal's Law in fluid mechanics.
Pascal's Law states that the pressure exerted at any point on a confined fluid is transmitted undiminished in all directions and acts with equal force on equal areas.
In this case, the piston on the left has a force acting on it due to its weight, which is given by the equation:
Force = mass x acceleration due to gravity (F = mg)
We can calculate the force acting on the piston using:
Force = mass x acceleration due to gravity (F = mg)
Now, let's calculate the force acting on the piston on the left:
Given:
Mass of the piston = 3.0 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Force = 3.0 kg x 9.8 m/s^2
Force = 29.4 N
Since the cross-sectional area of the piston is related to its diameter, we can calculate the area using the formula:
Area = π * (radius)^2
Given:
Diameter of the piston = 4.5 cm = 0.045 m
Radius = Diameter / 2 = 0.045 m / 2 = 0.0225 m
Area = π * (0.0225 m)^2
Area ≈ 0.0016 m^2
Now, let's apply Pascal's Law to calculate the pressure:
The pressure exerted on the left piston is given by:
Pressure = Force / Area
Pressure = 29.4 N / 0.0016 m^2
Pressure ≈ 18375 Pa
Since the pressure is the same throughout the fluid in the system, the pressure on the right piston is also 18375 Pa.
Lastly, we can use the relationship between pressure, density, and height difference to calculate the height difference (h) between the two pistons:
Pressure = density x gravity x height difference (P = ρgh)
Rearranging the formula to solve for the height difference:
Height difference (h) = Pressure / (density x gravity)
Given:
Density of the fluid (ρ) = 710 kg/m^3
Gravity (g) = 9.8 m/s^2
Height difference (h) = 18375 Pa / (710 kg/m^3 x 9.8 m/s^2)
Height difference (h) ≈ 2.52 m
Therefore, the height difference (h) between the two pistons is approximately 2.52 m.