A hydrolic press consist of a pum, load and two cylinders in which the large sylinder is eight times the diameter of smaller one. Use a well labelled cross_sectional area diagramdiagram of hydrolic press to determine the value of mechanical advantage of this machine if it is working with efficiency of 90%
To determine the mechanical advantage of the hydraulic press, we need to understand the hydraulic system's structure. It consists of a pump, load, and the two cylinders, with the larger cylinder being eight times the diameter of the smaller one.
Let's label the cross-sectional areas of the two cylinders as follows:
- Smaller Cylinder: A1
- Larger Cylinder: A2
As we know, the cross-sectional area of a cylinder is directly proportional to the square of its diameter. Given that the larger cylinder's diameter is eight times that of the smaller one, we can express the relationship between their cross-sectional areas as:
A2 = (8)^2 * A1
A2 = 64 * A1
Now, let's label the pressure in the smaller cylinder as P1 and the pressure in the larger cylinder as P2.
According to Pascal's principle, the pressure in a liquid is equal in all directions, and pressure is given by the formula:
Pressure = Force / Area
Since the pressure in the liquid is equal, we can equate the forces acting on the two cylinders:
Force1 = Force2
Using the pressure formula, we can express the forces in terms of pressure and area:
P1 * A1 = P2 * A2
Substituting the relationship between the areas, we get:
P1 * A1 = P2 * 64 * A1
Simplifying:
P1 = 64 * P2
Finally, let's define the mechanical advantage (MA) as the ratio of the force output (F2) to the force input (F1):
MA = F2 / F1
Since the force is directly proportional to the pressure and area, we can express it as:
F1 = P1 * A1
F2 = P2 * A2
Substituting the relationship between pressures:
F1 = (64 * P2) * A1
F2 = P2 * 64 * A1
Calculating the mechanical advantage:
MA = (P2 * 64 * A1) / (64 * P2 * A1)
MA = 1
Therefore, the mechanical advantage of this hydraulic press is 1.
Note: The efficiency of the machine does not affect the calculation of mechanical advantage. The efficiency refers to the ratio of useful work output to the work input, whereas mechanical advantage is solely concerned with the force amplification.
To determine the mechanical advantage of a hydraulic press, we need to calculate the ratio of the output force to the input force. The mechanical advantage (MA) can be calculated using the following formula:
MA = (output force) / (input force)
In this case, the output force is the force exerted by the larger cylinder, and the input force is the force exerted by the smaller cylinder. Since the hydraulic press is working with an efficiency of 90%, we need to take this into account as well.
First, let's consider the cross-sectional areas of the cylinders. Since the large cylinder is eight times the diameter of the smaller one, the ratio of their areas is the square of that ratio, which is 64:1.
Let's label the cross-sectional area of the smaller cylinder as A1 and the cross-sectional area of the larger cylinder as A2. We can set up the equation:
A2 / A1 = 64 / 1
Since A2 is eight times the diameter of A1 (or four times the radius), we can rewrite the equation as:
(π*r2^2) / (π*r1^2) = 64 / 1
Simplifying the equation, we get:
r2^2 / r1^2 = 64 / 1
Taking the square root of both sides, we get:
r2 / r1 = 8 / 1
Now, let's consider the forces exerted by the cylinders. The force exerted by a cylinder is equal to the pressure multiplied by the cross-sectional area. Let's label the pressure exerted by the smaller cylinder as P1 and the pressure exerted by the larger cylinder as P2.
The output force (F2) is equal to P2 multiplied by A2, and the input force (F1) is equal to P1 multiplied by A1.
Now, taking into account the efficiency of 90%, the formula for mechanical advantage becomes:
MA = (F2 * 0.9) / F1
Substituting the expressions for F2 and F1:
MA = (P2 * A2 * 0.9) / (P1 * A1)
Substituting the expressions for A2 and A1:
MA = (P2 * 64 * 0.9) / (P1 * 1)
The cross-sectional area ratio of 64:1 cancels out, and we get:
MA = (P2 * 0.9) / P1
Therefore, the mechanical advantage of the hydraulic press with a 90% efficiency is given by the ratio of the pressure exerted by the larger cylinder (P2) to the pressure exerted by the smaller cylinder (P1) multiplied by 0.9.
To determine the mechanical advantage of the hydraulic press, we need to understand the concept of mechanical advantage first. Mechanical advantage is the ratio of the force output to the force input of a machine.
In a hydraulic press, the principle is based on Pascal's law, which states that when pressure is applied to a fluid in a confined space, it is transmitted equally in all directions.
Now, let's proceed step by step to determine the mechanical advantage:
Step 1: Draw a well-labeled cross-sectional area diagram of the hydraulic press.
______ ______
| | | |
| Pump | | Load |
|______| |______|
| |
_|___ ___|_
| | | |
| C1 | | C2 |
|_____| |_____|
Here, C1 represents the smaller cylinder with diameter D1, and C2 represents the larger cylinder with diameter D2. The force applied on the smaller cylinder is F1, and the force output or load is F2.
Step 2: Write down the relationship between force and cross-sectional area.
The force applied or load (F) is directly proportional to the cross-sectional area (A) of the cylinder. Mathematically, F = PA, where P is the pressure.
The pressure (P) is the same in both cylinders due to Pascal's law.
Step 3: Express the relationship between the forces and areas.
For the smaller cylinder (C1):
F1 = P * A1
For the larger cylinder (C2):
F2 = P * A2
Step 4: Find the relationship between the areas.
The area of a circle is proportional to the square of its diameter.
A1 = π * (D1/2)^2
A2 = π * (D2/2)^2
Given that the diameter of the larger cylinder (D2) is eight times the diameter of the smaller cylinder (D1):
D2 = 8 * D1
Substituting the values of A1 and A2:
A1 = π * (D1/2)^2
A2 = π * [(8 * D1)/2]^2
Simplifying the equations:
A1 = π * (D1/2)^2 = π * (D1^2 / 4)
A2 = π * [(8 * D1)/2]^2 = π * (16 * D1^2)
Step 5: Find the ratio between the forces.
The mechanical advantage (MA) of the hydraulic press is given by the ratio of the output force (F2) to the input force (F1).
MA = F2 / F1
Substituting the values of forces:
MA = (P * A2) / (P * A1)
Since the pressure (P) is the same in both cylinders, it cancels out:
MA = A2 / A1
Step 6: Simplify the equation.
Substituting the values of A1 and A2:
MA = (π * (16 * D1^2)) / (π * (D1^2 / 4))
Canceling out π:
MA = (16 * D1^2) / (D1^2 / 4)
Simplifying further:
MA = (16 * D1^2) * (4 / D1^2)
Canceling out D1^2:
MA = 16 * 4
MA = 64
Step 7: Calculate the efficiency-adjusted mechanical advantage.
Since the efficiency of the machine is 90%, the actual mechanical advantage is equal to the mechanical advantage multiplied by the efficiency.
Efficiency-adjusted MA = 64 * 0.9 = 57.6
Therefore, the value of the mechanical advantage of this hydraulic press, when working with an efficiency of 90%, is 57.6.