Hydraulic Force Calculation Using Piston Radii

This problem involves calculating the input force required for a hydraulic system based on the principles of fluid mechanics, specifically Pascal's principle.

Identifying Given Information

• Input Piston Radius, $r_1 = 2$ cm
• Output Piston Radius, $r_2 = 8$ cm
• Output Force, $F_2 = 8000$ N
• Input Force, $F_1 = ?$

Applying Pascal's Principle

Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. In a hydraulic system with two pistons, this leads to the relationship:

$ \frac{F_1}{A_1} = \frac{F_2}{A_2} $

Where $F_1$ and $F_2$ are the forces on the input and output pistons, respectively, and $A_1$ and $A_2$ are their areas.

Since the area of a piston is given by $A = \pi r^2$, the formula can be rewritten in terms of radii:

$ \frac{F_1}{\pi r_1^2} = \frac{F_2}{\pi r_2^2} $

The $\pi$ terms cancel out, simplifying the equation to:

$ \frac{F_1}{r_1^2} = \frac{F_2}{r_2^2} $

Calculating the Input Force ($F_1$)

Rearranging the formula to solve for $F_1$:

$ F_1 = F_2 \times \frac{r_1^2}{r_2^2} $

Now, substitute the given values:

$ F_1 = 8000 \, \text{N} \times \frac{(2 \, \text{cm})^2}{(8 \, \text{cm})^2} $

$ F_1 = 8000 \, \text{N} \times \frac{4 \, \text{cm}^2}{64 \, \text{cm}^2} $

$ F_1 = 8000 \, \text{N} \times \frac{1}{16} $

$ F_1 = 500 \, \text{N} $

Conclusion

The force needed to apply to the smaller piston is 500 N.