A rectangular block has a square base measuring `axxa` and its height is `h`. It moves on a horizontal surface in a direction perpendicular to one of the edges. The coefficient of friction is `mu`. It will topple if

Options

A. `mu gt h/a`
B. `mugta/h`
C. `mugt(2a)/h`
D. `mugta/(2h)`

Explanation

To determine the condition under which a rectangular block with a square base will topple when moving on a horizontal surface, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Block**:
- The gravitational force (weight) acting downward, \( W = mg \).
- The normal force \( N \) acting upward, which balances the weight, so \( N = mg \).
- The frictional force \( F \) acting in the direction opposite to the motion, given by \( F = \mu N = \mu mg \).

2. **Understand the Toppling Condition**:
- The block will start to topple when the torque due to the frictional force about the pivot point (the edge of the base) exceeds the torque due to the weight of the block acting through its center of mass.

3. **Calculate the Torque Due to Friction**:
- The torque \( \tau_f \) due to the frictional force about the pivot point is given by:
\[ \tau_f = F \cdot \frac{h}{2} = \mu mg \cdot \frac{h}{2} \]

4. **Calculate the Torque Due to Weight**:
- The torque \( \tau_w \) due to the weight of the block about the pivot point is given by:
\[ \tau_w = mg \cdot \frac{a}{2} \]

5. **Set Up the Condition for Toppling**:
- The block will start to topple when the torque due to friction equals the torque due to the weight:
\[ \tau_f > \tau_w \]
- Thus, we have:
\[ \mu mg \cdot \frac{h}{2} > mg \cdot \frac{a}{2} \]

6. **Simplify the Equation**:
- Cancel \( mg \) from both sides:
\[ \mu \cdot \frac{h}{2} > \frac{a}{2} \]
- This simplifies to:
\[ \mu h > a \]

7. **Final Condition for Toppling**:
- Rearranging gives:
\[ \mu > \frac{a}{h} \]

### Conclusion:
The rectangular block will start to topple if the coefficient of friction \( \mu \) is greater than the ratio of the base side length \( a \) to the height \( h \):
\[ \mu > \frac{a}{h} \]