Motion of an oscillating liquid column in a U-tube is

Options

A. periodic but not simple harmonic.
B. non-periodic
C. simple harmonic and time period is independent of the density of the liquid.
D. simple harmonic and time period is directly proportional to the density of the liquid.

Explanation

To solve the question about the motion of an oscillating liquid column in a U-tube, we can follow these steps:

### Step 1: Understanding the Motion
The motion of a liquid column in a U-tube can be analyzed by considering the forces acting on the liquid due to gravity and the restoring force that arises when the liquid is displaced from its equilibrium position.

**Hint:** Consider how the forces change when the liquid is displaced.

### Step 2: Identifying the Type of Motion
The oscillation of the liquid column can be classified based on its characteristics. In this case, we need to determine if the motion is periodic and if it can be classified as simple harmonic motion (SHM).

**Hint:** Recall the definitions of periodic motion and simple harmonic motion.

### Step 3: Deriving the Time Period
For a U-tube, the time period \( T \) of oscillation can be derived using the formula:

\[ T = 2\pi \sqrt{\frac{h}{g}} \]

where:
- \( T \) is the time period,
- \( h \) is the height of the liquid column,
- \( g \) is the acceleration due to gravity.

**Hint:** Think about how the height of the liquid column affects the time period.

### Step 4: Analyzing the Dependence on Density
From the derived formula, we can see that the time period \( T \) does not include the density of the liquid. This means that the time period is independent of the density of the liquid.

**Hint:** Check if density appears in the formula for time period.

### Step 5: Conclusion
Based on the analysis, we can conclude that the motion of the oscillating liquid column in a U-tube is simple harmonic and the time period is independent of the density of the liquid.

**Final Answer:** The correct option is that the motion is simple harmonic and the time period is independent of the density of the liquid.

### Summary of the Steps:
1. Understand the forces acting on the liquid.
2. Identify the type of motion (periodic vs. simple harmonic).
3. Derive the time period using the appropriate formula.
4. Analyze the dependence of the time period on the density.
5. Conclude based on the findings.