When a second student joins the first, the piston sinks. What is the second student's mass?

Verified step by step guidance

1. Step 1: Understand the problem setup. The system involves a hydraulic lift with an incompressible fluid. The car is on one side of the piston, and the students are on the other side. The mass of the car is 1450 kg, and the mass of the first student is 85 kg. We need to find the mass of the second student when the piston sinks due to their combined weight.
2. Step 2: Recall Pascal's principle, which states that pressure is transmitted equally throughout the incompressible fluid. The pressure exerted by the car on one side of the piston must equal the pressure exerted by the students on the other side.
3. Step 3: Write the pressure equation. Pressure is defined as force divided by area: \( P = \frac{F}{A} \). The force exerted by the car is \( F_{car} = m_{car} \cdot g \), and the force exerted by the students is \( F_{students} = (m_{student1} + m_{student2}) \cdot g \). The areas of the pistons are proportional to their diameters.
4. Step 4: Set up the equality of pressures: \( \frac{F_{car}}{A_{car}} = \frac{F_{students}}{A_{students}} \). Substitute \( F_{car} = m_{car} \cdot g \) and \( F_{students} = (m_{student1} + m_{student2}) \cdot g \). The gravitational acceleration \( g \) cancels out, leaving \( \frac{m_{car}}{A_{car}} = \frac{m_{student1} + m_{student2}}{A_{students}} \).
5. Step 5: Solve for \( m_{student2} \). The areas \( A \) are proportional to the square of the diameters of the pistons. Use the given dimensions to calculate the ratio of the areas, and rearrange the equation to isolate \( m_{student2} \): \( m_{student2} = \frac{m_{car} \cdot A_{students}}{A_{car}} - m_{student1} \). Substitute the known values to find the mass of the second student.

Key Concepts

Buoyancy

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This principle, described by Archimedes, states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. In this scenario, the piston sinks due to the combined weight of the car and the second student, which affects the fluid displacement.

Weight and Mass

Weight is the force exerted by gravity on an object, calculated as the product of mass and gravitational acceleration (W = mg). In this context, the mass of the second student contributes to the total weight acting on the piston, influencing how much the piston sinks into the incompressible fluid. Understanding the relationship between mass and weight is crucial for solving the problem.

Incompressible Fluid

An incompressible fluid is one that does not change in volume when subjected to pressure. In this problem, the fluid's incompressibility means that any additional weight (from the second student) will directly affect the fluid's pressure and the buoyant force acting on the piston. This concept is essential for analyzing how the system behaves when the second student joins the first.