Question

A hydraulic lift is to be used to lift a 2500 kg weight by putting a weight of \(25 \mathrm{kg}\) on a piston with a diameter of \(10 \mathrm{cm} .\) Determine the diameter of the piston on which the weight is to be placed.

Short answer

Answer: The diameter of the piston on which the 2500 kg weight is to be placed is approximately 40 cm.

Step-by-step solution

Calculate the area of the first piston

We are given the diameter of the first piston which is 10 cm. To find the area of this piston, we can use the formula for the area of a circle: \(A_1 = \frac{\pi d_1^2}{4}\) Plugging in the given diameter: \(A_1 = \frac{\pi (10)^2}{4} = 25 \pi \; \mathrm{cm}^2\)

Apply Pascal's Law

Pascal's Law states that the pressure applied in a closed fluid at rest is transmitted evenly throughout the fluid, and can be written as \(F_1/A_1 = F_2/A_2\). We already have the area of the first piston (A_1), and the forces applied to each piston can be found by multiplying their weights by the acceleration due to gravity (g = 9.81 m/s²). \(F_1 = m_1 g = 25 \mathrm{kg} \times 9.81 \frac{\mathrm{m}}{\mathrm{s}^2} = 245.25 \mathrm{N}\) \(F_2 = m_2 g = 2500 \mathrm{kg} \times 9.81 \frac{\mathrm{m}}{\mathrm{s}^2} = 24525 \mathrm{N}\) Now, we can plug these forces and the area of the first piston into Pascal's Law: \(\frac{F_1}{A_1} = \frac{F_2}{A_2}\)

Solve for the area of the second piston

Now that we have the forces and the area of the first piston, we can solve for the area of the second piston using Pascal's Law: \(\frac{245.25 \mathrm{N}}{25 \pi \mathrm{cm}^2} = \frac{24525 \mathrm{N}}{A_2}\) Cross-multiply and solve for \(A_2\): \(A_2 = \frac{24525 \mathrm{N} \times 25 \pi \mathrm{cm}^2}{245.25 \mathrm{N}} \approx 1000 \pi \mathrm{cm}^2\)

Find the diameter of the second piston

Now that we have the area of the second piston, we can find its diameter using the formula for the area of a circle: \(A_2 = \frac{\pi d_2^2}{4}\) Solve for the diameter: \(d_2^2 = \frac{4A_2}{\pi} \Rightarrow d_2 = \sqrt{\frac{4 \times 1000 \pi \mathrm{cm}^2}{\pi}} \Rightarrow d_2 \approx 40\mathrm{cm}\) The diameter of the piston on which the weight is to be placed is approximately 40 cm.