the small piston of a hydraulic press has an area of 7.0 cm squared. if the applied force is 30.0 N, find the area of the large piston to exert a pressing force of 4000 N
4000/30 = 133.3 times the force requires 133.3 times larger area for the large piston, compared to the smaller one.
7.0 cm^2 * 133.3 = ?
To find the area of the large piston, we can use the principle of Pascal's Law, which states that pressure is transmitted equally in all directions in a fluid.
The formula relating force, pressure, and area is:
Force = Pressure * Area
First, we'll find the pressure exerted by the small piston:
Pressure = Force / Area
Given that the applied force is 30.0 N and the area of the small piston is 7.0 cm²:
Pressure = 30.0 N / 7.0 cm²
Next, we can use the pressure from the small piston to find the area of the large piston:
Area_large = Force_large / Pressure
Given that the pressing force is 4000 N and we have already calculated the pressure from the small piston:
Area_large = 4000 N / Pressure
Now, let's calculate the area of the large piston:
Area_large = 4000 N / (30.0 N / 7.0 cm²)
Simplifying the equation:
Area_large = (4000 N * 7.0 cm²) / 30.0 N
Area_large = 933.33 cm²
Therefore, the area of the large piston needed to exert a pressing force of 4000 N is approximately 933.33 cm².
To find the area of the large piston, we can use the principle of Pascal's law in hydraulics. Pascal's law states that the pressure applied to an enclosed fluid is transmitted uniformly in all directions.
1. First, let's calculate the pressure exerted by the small piston. Pressure is defined as force per unit area.
Pressure = Force / Area
Plugging in the values:
Pressure = 30.0 N / 7.0 cm²
2. We need to convert the area from cm² to m², as the SI unit for pressure is Pascal (Pa), which is equal to N/m².
1 cm² = (1 cm * 1 cm) = (0.01 m * 0.01 m) = 0.0001 m²
Therefore, Area = 7.0 cm² * 0.0001 m²/cm²
3. Now, we can calculate the pressure exerted by the small piston:
Pressure = 30.0 N / (7.0 cm² * 0.0001 m²/cm²)
4. The pressure exerted by the small piston will be the same as the pressure exerted by the large piston, as per Pascal's law.
Pressure = Force / Area
Plugging in the values:
4000 N = Pressure * Area_large
5. Rearranging the equation to solve for the area of the large piston:
Area_large = 4000 N / Pressure
6. Substitute the value of the pressure we calculated in step 3 into the equation:
Area_large = 4000 N / (30.0 N / (7.0 cm² * 0.0001 m²/cm²))
7. Simplifying the equation:
Area_large = 4000 N / (30.0 N / (0.0007 m²))
Area_large = 4000 N * (0.00007 m² / 30.0 N)
Area_large = (14/1000) m²
8. Finally, convert the area to cm² to get the final answer:
Area_large = (14/1000) m² * (10000 cm²/m²)
Area_large = 140 cm²
Therefore, the area of the large piston required to exert a pressing force of 4000 N is 140 cm².