Question

In a hydraulic press, the piston in the small cylinder has a diameter of \(d_{1}\) cm, while the larger piston has a diameter of \(d_{2}\) cm. If a force \(F_{1}\) is applied to the small piston, then the force \(F_{2}\) on the large piston is given by

• A. \(F_{2}=\frac{d_{2}^{2}}{d_{1}^{2}}F_{1}\)
• B. \(F_{2}=\frac{d_{1}^{2}}{d_{2}^{2}}F_{1}\)
• C. \(F_{2}=\frac{d_{2}^{2}}{d_{1}^{2}}\frac{1}{F_{1}}\)
• D. \(F_{2}=\frac{d_{1}^{2}}{d_{2}^{2}}\frac{1}{F_{1}}\)

Hint:

Pascal's law says the pressure is the same on both pistons, so force is proportional to piston area, which varies as diameter squared.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
By Pascal's law, the pressure applied on the small piston is transmitted equally to the large piston: \(\frac{F_{1}}{A_{1}}=\frac{F_{2}}{A_{2}}\).

Step 2: Areas
\(A_{1}=\frac{\pi d_{1}^{2}}{4}\) and \(A_{2}=\frac{\pi d_{2}^{2}}{4}\).

Step 3: Calculation
\(F_{2}=F_{1}\frac{A_{2}}{A_{1}}=\frac{d_{2}^{2}}{d_{1}^{2}}F_{1}\).

Step 4: Compare the options
This is option (A).

Final Answer:
The force on the large piston is \(\frac{d_{2}^{2}}{d_{1}^{2}}F_{1}\). \[ \boxed{\text{(A) }F_{2}=\dfrac{d_{2}^{2}}{d_{1}^{2}}F_{1}} \]