A solid cylinder of radius `m` is kept on an inclined plane on its base and it does not slide. The angle of incline plane is `53^(@)` with horizontal and it is fixed. What will be maxcimum height (in meters) of the cylinder for it not to topple.

Explanation

To solve the problem of finding the maximum height of a solid cylinder on an inclined plane without toppling, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have a solid cylinder of radius \( r \) placed on an inclined plane that makes an angle of \( 53^\circ \) with the horizontal.
- The cylinder will not slide, and we need to determine the maximum height \( h \) for it not to topple.

2. **Identifying the Angle**:
- The angle of the inclined plane is \( 53^\circ \).
- The complementary angle (the angle between the vertical and the inclined plane) is \( 90^\circ - 53^\circ = 37^\circ \).

3. **Setting Up the Right Triangle**:
- When the cylinder is about to topple, the center of mass will be vertically above the edge of the base in contact with the inclined plane.
- We can visualize a right triangle where:
- The height from the base of the cylinder to the center of mass is \( \frac{h}{2} \).
- The horizontal distance from the edge of the base to the vertical line through the center of mass is \( r \) (the radius of the cylinder).

4. **Using Trigonometry**:
- In the right triangle formed, we can use the tangent of the angle \( 37^\circ \):
\[ \tan(37^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\frac{h}{2}}{r} \]
- Rearranging gives:
\[ h = 2r \tan(37^\circ) \]

5. **Calculating \( \tan(37^\circ) \)**:
- The value of \( \tan(37^\circ) \) is approximately \( \frac{3}{4} \) or \( 0.75 \).

6. **Substituting Values**:
- Now substituting \( r = 2 \, m \) (as given in the problem):
\[ h = 2 \times 2 \times \tan(37^\circ) = 2 \times 2 \times \frac{3}{4} \]
- Simplifying this:
\[ h = 4 \times \frac{3}{4} = 3 \, m \]

7. **Final Answer**:
- Therefore, the maximum height \( h \) of the cylinder for it not to topple is \( 3 \, m \).

### Summary:
The maximum height of the cylinder for it not to topple is \( 3 \, m \).