Consider a thin square plate floating on a viscolas liquid in a large tank. The heighth of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity `u_(0).` Which of the following statements is (are) true?

Options

A. The resistive force of liquid on the plate is inversely proportional to h
B. The resistive force of liquid on the plate is independent of the area of the plate
C. The tangential (shear) stress on the floor of the tank increases with `u_(0)`
D. The tangential (shear) stress on the plate varies linearly with the viscosity `eta` of the liquid

Explanation

To solve the problem, we will analyze the situation step by step based on the principles of fluid mechanics, particularly focusing on Newton's law of viscosity.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a thin square plate floating on a viscous liquid in a large tank.
- The height of the liquid (H) is much less than the width of the tank.
- The plate is being pulled horizontally with a constant velocity \( u_0 \).

2. **Identify the Forces Acting on the Plate**:
- As the plate moves, it experiences a resistive viscous force \( F_v \) from the liquid, which opposes its motion.

3. **Apply Newton's Law of Viscosity**:
- According to Newton's law of viscosity, the viscous force can be expressed as:
\[ F_v = \mu \cdot A \cdot \frac{\Delta v}{\Delta y} \]
- Where:
- \( \mu \) is the dynamic viscosity of the fluid,
- \( A \) is the area of the plate,
- \( \Delta v \) is the change in velocity (from 0 at the bottom layer to \( u_0 \) at the plate),
- \( \Delta y \) is the distance between the plate and the bottom layer (which is \( h \)).

4. **Calculate the Viscous Force**:
- From the above equation, substituting \( \Delta v = u_0 \) and \( \Delta y = h \):
\[ F_v = \mu \cdot A \cdot \frac{u_0}{h} \]

5. **Analyze the Statements**:
- **Statement 1**: The resistive force of the liquid on the plate is inversely proportional to \( h \).
- True, as seen from the formula \( F_v \propto \frac{1}{h} \).
- **Statement 2**: The resistive force of the liquid on the plate is independent of the area of the plate.
- False, as \( F_v \) is directly proportional to the area \( A \).
- **Statement 3**: The tangential stress on the floor of the tank increases with \( u_0 \).
- True, as tangential stress \( \tau = \frac{F_v}{A} = \frac{\mu \cdot u_0}{h} \) is directly proportional to \( u_0 \).
- **Statement 4**: The tangential or shear stress on the plate varies linearly with viscosity \( \eta \) of the fluid.
- True, as \( \tau \) is directly proportional to \( \mu \).

6. **Conclusion**:
- The true statements are:
- Statement 1: True
- Statement 2: False
- Statement 3: True
- Statement 4: True

### Final Answer:
The true statements are 1, 3, and 4.