Hydraulic Lift II.The piston of a hydraulic automobile lift is 0.30 m in diameter. What gauge pressure, in pascals, is required to lift a car with a mass of 1200 kg? Also express this pressure in atmospheres.

Verified step by step guidance

1. First, calculate the area of the piston using the formula for the area of a circle: \( A = \pi r^2 \). The radius \( r \) is half of the diameter, so \( r = 0.30 \text{ m} / 2 = 0.15 \text{ m} \). Substitute this value into the formula to find the area.
2. Next, determine the force required to lift the car. The force \( F \) is equal to the weight of the car, which can be calculated using \( F = mg \), where \( m = 1200 \text{ kg} \) is the mass of the car and \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity.
3. Now, calculate the gauge pressure \( P \) required to lift the car using the formula \( P = \frac{F}{A} \). Substitute the values of \( F \) and \( A \) from the previous steps into this formula to find the pressure in pascals.
4. To convert the pressure from pascals to atmospheres, use the conversion factor: \( 1 \text{ atm} = 101325 \text{ Pa} \). Divide the pressure in pascals by 101325 to express it in atmospheres.
5. Review the calculations to ensure all units are consistent and the steps are followed correctly. This will give you the gauge pressure required in both pascals and atmospheres.

Key Concepts

Pascal's Principle

Pascal's Principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of its container. This principle is fundamental in hydraulic systems, where pressure applied at one point is transmitted to lift heavy objects, such as cars, using a hydraulic lift.

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It is the difference between absolute pressure and atmospheric pressure. In hydraulic systems, gauge pressure is used to determine the additional pressure needed to perform tasks, such as lifting a car, beyond what is exerted by the atmosphere.

Pressure Conversion

Pressure conversion involves changing pressure units from one system to another, such as from pascals to atmospheres. This is essential for understanding and communicating pressure values in different contexts, as atmospheric pressure is a common reference point in everyday situations.