Two-thirds of a container is filled with a liquid and then the cylinder is sealed. F is the force exerted at the bottom of the container and P is the pressure exerted at the surface. On removing some air from the container with the help of a vacuum pump

Options

A. Pand F both decrease
B. P and F both increase
C. The liquid level in container will fall
D. The liquid level in container will be unchanged

Answer

To solve the problem step by step, we will analyze the situation in the sealed cylindrical container filled with liquid and air, and how the removal of air affects the force and pressure.

### Step 1: Understand the Initial Conditions
- The cylindrical container is filled with liquid up to two-thirds of its height.
- Above the liquid, there is air.
- Let the pressure exerted by the air at the surface be \( P_a \).
- The pressure at the bottom of the container, where the liquid is, is due to the weight of the liquid column and the air pressure above it.

### Step 2: Calculate the Initial Pressure at the Bottom
- The pressure at the bottom of the container can be expressed as:
\[ P = P_a + \rho g h \]
where:
- \( P \) is the total pressure at the bottom,
- \( \rho \) is the density of the liquid,
- \( g \) is the acceleration due to gravity,
- \( h \) is the height of the liquid column.

### Step 3: Understand the Effect of Removing Air
- When air is removed from the container using a vacuum pump, the pressure \( P_a \) exerted by the air decreases.
- According to the ideal gas law, if the volume and temperature are constant, reducing the number of air molecules will decrease the air pressure.

### Step 4: Analyze the Change in Pressure
- As the air pressure \( P_a \) decreases, the total pressure at the bottom of the container becomes:
\[ P' = P_a' + \rho g h \]
where \( P_a' < P_a \).
- Therefore, the total pressure \( P' \) at the bottom of the container will also decrease.

### Step 5: Calculate the Force at the Bottom
- The force \( F \) exerted at the bottom of the container is given by:
\[ F = P \cdot A \]
where \( A \) is the area of the base of the cylinder.
- Since \( P \) decreases due to the decrease in \( P_a \), the force \( F \) will also decrease:
\[ F' = P' \cdot A \]

### Conclusion
- Both the pressure \( P \) at the bottom of the container and the force \( F \) exerted at the bottom will decrease when air is removed from the container.

### Summary of Results
1. The pressure \( P \) at the bottom of the container decreases.
2. The force \( F \) exerted at the bottom of the container also decreases.
3. The liquid level in the container remains unchanged because liquids are incompressible.