In a car lift , compressed air with a gauge pressure of `4.0 xx 10^(5)` Pa is used to raise a piston with a circular cross-sectional area . If the radius of the piston is 0.17 m , what is the maximum mass that can be raised using this piston ?

Options

A. 530 kg
B. 3700 kg
C. 9800 kg
D. 22,000 kg

Answer

To solve the problem of determining the maximum mass that can be raised using a piston in a car lift, we can follow these steps:

### Step 1: Understand the relationship between pressure, force, and area
According to Pascal's law, the pressure exerted on a confined fluid is transmitted undiminished throughout the fluid. The relationship between pressure (P), force (F), and area (A) is given by the formula:

\[ P = \frac{F}{A} \]

Where:
- \( P \) is the pressure in Pascals (Pa)
- \( F \) is the force in Newtons (N)
- \( A \) is the area in square meters (m²)

### Step 2: Identify the given values
From the problem, we have:
- Gauge pressure \( P_g = 4.0 \times 10^5 \, \text{Pa} \)
- Radius of the piston \( r = 0.17 \, \text{m} \)

### Step 3: Calculate the area of the piston
The area \( A \) of a circular piston can be calculated using the formula:

\[ A = \pi r^2 \]

Substituting the value of \( r \):

\[ A = \pi (0.17)^2 \]
\[ A \approx 3.14 \times 0.0289 \]
\[ A \approx 0.049 \, \text{m}^2 \]

### Step 4: Relate the force to mass
The force \( F \) exerted by the piston can also be expressed in terms of mass \( m \) and gravitational acceleration \( g \):

\[ F = mg \]

Where \( g \approx 9.8 \, \text{m/s}^2 \).

### Step 5: Set up the equation using pressure
Using the relationship between pressure, force, and area, we can substitute \( F \) with \( mg \):

\[ P_g = \frac{mg}{A} \]

Rearranging this gives:

\[ m = \frac{P_g \cdot A}{g} \]

### Step 6: Substitute the values into the equation
Now we can substitute \( P_g \), \( A \), and \( g \) into the equation:

\[ m = \frac{(4.0 \times 10^5) \cdot (0.049)}{9.8} \]

Calculating this step-by-step:

1. Calculate \( P_g \cdot A \):
\[ P_g \cdot A = 4.0 \times 10^5 \times 0.049 \approx 19600 \, \text{N} \]

2. Now divide by \( g \):
\[ m \approx \frac{19600}{9.8} \approx 2000 \, \text{kg} \]

### Step 7: Final calculation
After calculating, we find:

\[ m \approx 2000 \, \text{kg} \]

### Conclusion
The maximum mass that can be raised using the piston is approximately **2000 kg**.

---