Question

In a hydraulic lift, the surface area of the input piston is 6 cm² and that of the output piston is 1500 cm². If 100 N force is applied to the input piston to raise the output piston by 20 cm, then the work done is kJ.

Answer 1

Correct Answer: 5

To determine the work done, we start with Pascal's principle, which states that the pressure applied to a confined fluid is transmitted undiminished throughout the fluid.

Given:

• Surface area of input piston, A₁ = 6 cm²
• Surface area of output piston, A₂ = 1500 cm²
• Force on input piston, F₁ = 100 N
• Output piston raised by h = 20 cm = 0.2 m

The pressure applied at the input piston is:

P = F₁ / A₁ = 100 N / 6 cm²

Since 1 cm² = 0.0001 m²:

A₁ = 6 × 0.0001 = 0.0006 m²

P = 100 N / 0.0006 m² = 166666.67 N/m²

This pressure is transmitted to the output piston:

F₂ = P × A₂ = 166666.67 N/m² × (1500 cm² × 0.0001 m²/cm²) = 25000 N

The work done, W, is given by:

W = F₂ × h = 25000 N × 0.2 m = 5000 J

Converting 5000 J to kJ:

W = 5 kJ

The work done is 5 kJ.

Answer 2

Step 1: Given data.
Area of input piston, \( A_1 = 6 \, \text{cm}^2 \)
Area of output piston, \( A_2 = 1500 \, \text{cm}^2 \)
Force applied on input piston, \( F_1 = 100 \, \text{N} \)
Displacement of output piston, \( h_2 = 20 \, \text{cm} = 0.2 \, \text{m} \)

Step 2: Apply Pascal’s law.
According to Pascal’s law, the pressure is transmitted equally in all directions in an enclosed fluid. Therefore,
\[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \] Hence, the force on the output piston is:
\[ F_2 = F_1 \times \frac{A_2}{A_1} \] Substitute the given values:
\[ F_2 = 100 \times \frac{1500}{6} = 100 \times 250 = 25000 \, \text{N} \]

Step 3: Relation between displacements.
Since the volume of the fluid displaced remains the same,
\[ A_1 h_1 = A_2 h_2 \] Therefore, \[ h_1 = \frac{A_2 h_2}{A_1} = \frac{1500 \times 0.2}{6} = 50 \, \text{m} \]

Step 4: Calculate the work done.
Work done on the input side = \( F_1 \times h_1 \)
\[ W = 100 \times 50 = 5000 \, \text{J} = 5 \, \text{kJ} \]

Final Answer:
\[ \boxed{5 \, \text{kJ}} \]