A hydraulic press with the larger piston of diameter 35 cm at a heigth of 1.5 m relative to the smaller piston of diameter 10 cm. The mass on the smaller piston is 20 kg. What is the force exerted on the load placed. On the larger piston ? The density of oil in the press is `750 kg m^(-3)`

Explanation

To solve the problem step by step, we will use the principles of hydraulics and the relationship between pressure, force, and area.

### Step 1: Calculate the Area of the Smaller Piston
The area \( A_1 \) of the smaller piston can be calculated using the formula for the area of a circle:
\[ A_1 = \pi \left( \frac{D_1}{2} \right)^2 \]
Where \( D_1 = 10 \, \text{cm} = 0.1 \, \text{m} \).

Calculating:
\[ A_1 = \pi \left( \frac{0.1}{2} \right)^2 = \pi \left( 0.05 \right)^2 = \pi \times 0.0025 \approx 0.00785 \, \text{m}^2 \]

### Step 2: Calculate the Force on the Smaller Piston
The force \( F_1 \) exerted by the mass on the smaller piston is given by:
\[ F_1 = m \cdot g \]
Where \( m = 20 \, \text{kg} \) and \( g \approx 10 \, \text{m/s}^2 \).

Calculating:
\[ F_1 = 20 \cdot 10 = 200 \, \text{N} \]

### Step 3: Calculate the Area of the Larger Piston
The area \( A_2 \) of the larger piston can be calculated similarly:
\[ A_2 = \pi \left( \frac{D_2}{2} \right)^2 \]
Where \( D_2 = 35 \, \text{cm} = 0.35 \, \text{m} \).

Calculating:
\[ A_2 = \pi \left( \frac{0.35}{2} \right)^2 = \pi \left( 0.175 \right)^2 = \pi \times 0.030625 \approx 0.0962 \, \text{m}^2 \]

### Step 4: Calculate the Pressure at the Smaller Piston
The pressure \( P \) at the smaller piston is given by:
\[ P = \frac{F_1}{A_1} \]

Calculating:
\[ P = \frac{200}{0.00785} \approx 25464.5 \, \text{Pa} \]

### Step 5: Calculate the Pressure at the Larger Piston
The pressure at the larger piston must account for the height difference due to the oil. The pressure at the larger piston \( P_2 \) is given by:
\[ P_2 = P + \rho g h \]
Where \( \rho = 750 \, \text{kg/m}^3 \), \( g = 10 \, \text{m/s}^2 \), and \( h = 1.5 \, \text{m} \).

Calculating:
\[ P_2 = 25464.5 + (750 \cdot 10 \cdot 1.5) = 25464.5 + 11250 = 36714.5 \, \text{Pa} \]

### Step 6: Calculate the Force on the Larger Piston
Now we can find the force \( F_2 \) exerted on the larger piston using the pressure:
\[ F_2 = P_2 \cdot A_2 \]

Calculating:
\[ F_2 = 36714.5 \cdot 0.0962 \approx 3525.2 \, \text{N} \]

### Final Answer
The force exerted on the load placed on the larger piston is approximately \( 3525.2 \, \text{N} \).

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