Problem 24 - University Physics with Modern Physics

Question

The piston of a hydraulic automobile lift is 0.30 \(\mathrm{m}\) in diameter. What gauge pressure, in pascals, is required to lift a car with a mass of 1200 \(\mathrm{kg}\) ? Also express this pressure in atmospheres.

Step-by-Step Solution

Answer

166,500 Pa or 1.64 atm is needed to lift the car.

Step 1: Calculate the Area of the Piston

The diameter of the piston is given as 0.30 meters. First, we need to find the radius by dividing the diameter by 2, which gives \( r = \frac{0.30}{2} = 0.15 \) meters. The area \( A \) of a circle is calculated using the formula \( A = \pi r^2 \). Substitute the radius into the formula: \( A = \pi (0.15)^2 \approx 0.0707 \) square meters.

Step 2: Calculate the Force Needed to Lift the Car

The force needed to lift the car can be calculated using the gravitational force formula \( F = mg \), where \( m \) is the mass, and \( g \) is the acceleration due to gravity \( 9.81 \) m/s². So, \( F = 1200 \times 9.81 \approx 11772 \) newtons.

Step 3: Calculate the Gauge Pressure in Pascals

Gauge pressure can be calculated using the formula \( P = \frac{F}{A} \), where \( F \) is the force and \( A \) is the area of the piston. Substitute the values: \( P = \frac{11772}{0.0707} \approx 166500 \) pascals.

Step 4: Convert Pressure to Atmospheres

The pressure in pascals can be converted to atmospheres using the conversion factor 1 atm = 101325 pascals. Therefore, pressure in atmospheres \( = \frac{166500}{101325} \approx 1.64 \) atm.

Key Concepts

Gauge Pressure

When dealing with hydraulic systems like a hydraulic lift, gauge pressure is a vital concept. Gauge pressure measures the pressure in a system relative to the ambient atmospheric pressure. It is different from absolute pressure, which includes atmospheric pressure in its measurement. In our exercise, the gauge pressure is the pressure "over and above" the atmospheric pressure needed to lift the car.

To find the gauge pressure needed for the lift, we use the formula:

• \( P = \frac{F}{A} \)

This equation shows that pressure (\(P\)) is equal to the force (\(F\)) divided by the area (\(A\)) over which the force is applied. The solution calculates this as approximately 166,500 pascals. This is the pressure the lift system must exert to counteract the weight of the car and lift it from its initial state.

Piston Area Calculation

The area of the piston's circular surface is crucial in determining the pressure required for lifting. We need to calculate the area based on the diameter provided. For a circular piston, the area \( A \) is given by:

• \( A = \pi r^2 \)

where \( r \) is the radius of the circle. Since the diameter is 0.30 meters, we first halve it to find the radius \( r = 0.15 \) meters.

Substituting the radius into the area formula gives us:

• \( A = \pi (0.15)^2 \approx 0.0707 \text{ square meters} \)

This area calculation is fundamental because it sets the stage to calculate the gauge pressure in the previous section. Knowing the piston's area helps us understand how force is distributed in the hydraulic system and how much pressure is needed to lift the car.

Conversion to Atmospheres

Conversion of pressure from pascals to atmospheres is often needed for better understanding and comparison with everyday pressures. The pascal is the SI unit of pressure, but atmospheric pressure is commonly used in various fields and applications.

To convert the calculated pressure from pascals to atmospheres, we use the conversion factor:

• 1 atm = 101,325 pascals

Thus, the process is simple:

• \( \text{Pressure in atmospheres} = \frac{166,500 \text{ pascals}}{101,325 \text{ pascals/atm}} \)
• \( \approx 1.64 \text{ atm} \)

This conversion indicates that the gauge pressure required is about 1.64 times the pressure exerted by Earth's atmosphere at sea level. This perspective helps in visualizing the pressure level needed in the hydraulic system as it relates to familiar atmospheric conditions.