Explanation:

Metacentric height is a term used in naval architecture and fluid mechanics to describe the distance between the center of gravity of a floating body and its metacentre. The metacentre is the point at which a vertical line through the center of buoyancy of the body intersects with the body in the upright position.

When the metacentric height is increased, it means that the distance between the center of gravity and the metacentre is increased. This has a significant effect on the stability of the floating body and its time oscillation.

Effect of Increase in Metacentric Height:

1. Decreased Stability: When the metacentric height is increased, the stability of the floating body decreases. This is because the higher the metacentric height, the smaller the restoring moment acting on the body. In simple terms, the body becomes less stable and more prone to rolling or tilting.

2. Decreased Time Period of Oscillation: The time period of oscillation refers to the time taken for the floating body to complete one back-and-forth movement. When the metacentric height is increased, the time period of oscillation decreases. This is because the increased distance between the center of gravity and the metacentre reduces the restoring moment acting on the body, causing it to oscillate at a faster rate.

3. Increased Rolling Angle: The rolling angle refers to the angle of tilt or roll experienced by the floating body during oscillation. When the metacentric height is increased, the rolling angle also increases. This is because the increased distance between the center of gravity and the metacentre amplifies the effect of external forces, leading to larger rolling angles.

4. Decreased Damping: Damping refers to the resistance of a system to oscillation or movement. When the metacentric height is increased, the damping of the floating body decreases. This means that the body experiences less resistance to its oscillation, resulting in longer and more pronounced oscillations.

Conclusion:

In conclusion, when the metacentric height of a floating body is increased, its time oscillation is lower. This is due to the decreased stability, reduced time period of oscillation, increased rolling angle, and decreased damping. It is important to consider the metacentric height when designing and analyzing the stability and behavior of floating bodies.

Effect of Metacentric Height on Time Oscillation of a Floating Body

Metacentric height is a critical parameter that determines the stability of a floating body. It is the distance between the center of gravity (G) and the metacenter (M) of the body. The metacenter is the intersection point of the line of action of the buoyant force with the body in its initial equilibrium position.

1. Definition of Metacentric Height
Metacentric height (GM) can be mathematically expressed as GM = BM - BG, where BM is the distance between the metacenter (M) and the center of buoyancy (B), and BG is the distance between the center of buoyancy (B) and the center of gravity (G) of the floating body.

2. Significance of Metacentric Height
Metacentric height plays a crucial role in determining the stability of a floating body. It affects the body's ability to return to its equilibrium position after it is tilted or disturbed. A higher metacentric height indicates greater stability, while a lower metacentric height implies reduced stability.

3. Time Oscillation of a Floating Body
When a floating body is tilted from its equilibrium position, it experiences a restoring moment due to the buoyant force acting at a new position. This restoring moment acts to bring the body back to its original position. The time taken for the body to complete one oscillation is known as the time oscillation.

4. Effect of Metacentric Height on Time Oscillation
The time oscillation of a floating body is inversely proportional to the square root of the metacentric height. In other words, as the metacentric height decreases, the time oscillation increases.

When the metacentric height is reduced, it indicates that the center of gravity of the body is positioned lower compared to the metacenter. This results in a decreased restoring moment and a slower return to equilibrium. Therefore, the time taken for the body to complete one oscillation increases.

5. Answer Explanation
Based on the above explanation, option 'C' is correct. When the metacentric height increases, the time oscillation of a floating body will be lower. This is because a higher metacentric height provides greater stability and a stronger restoring moment, allowing the body to return to its equilibrium position more quickly. Conversely, a lower metacentric height reduces stability and prolongs the time taken for the body to complete one oscillation.