An empty upright cylinder is half filled with a liquid. How does adding the liquid affect the position of the combined centre of gravity of the cylinder–liquid system?

Introduction / Context:
Centre of gravity is an important concept in mechanics and stability. It is the point at which the entire weight of a body or system can be considered to act. This question considers an empty cylinder that is then half filled with a liquid. You need to predict how the position of the combined centre of gravity of the cylinder and the liquid changes compared to the original empty cylinder.

Given Data / Assumptions:

• An upright hollow cylinder is initially empty and standing on its base.
• The cylinder is uniform, so its own centre of gravity lies at mid height of its axis.
• The cylinder is then half filled with a liquid poured into it.
• The liquid has density greater than that of air and occupies the lower half of the cylinder.
• We consider the centre of gravity of the combined system of cylinder plus liquid.

Concept / Approach:
The combined centre of gravity of a system of bodies lies at the weighted average of their individual centres of gravity, with weights equal to their masses. Initially only the cylinder contributes, with its centre at mid height. After liquid is added, there is additional mass located in the lower part of the cylinder. The centre of gravity of the liquid in a half filled cylinder lies at the centre of the liquid column, which is at one quarter of the cylinder height from the base. The combined centre of gravity must lie between the cylinder centre and the liquid centre. Since the new mass is below the old centre, the overall centre of gravity shifts downward.

Step-by-Step Solution:
Step 1: Let the cylinder height be H. The centre of gravity of the empty cylinder is at H / 2 from the base. Step 2: When the cylinder is half filled, the liquid occupies height H / 2, so its centre of gravity is at H / 4 from the base. Step 3: Denote the mass of the cylinder as mc and the mass of the added liquid as ml. Step 4: The combined centre of gravity position y measured from the base is y = (mc * (H / 2) + ml * (H / 4)) / (mc + ml). Step 5: Since H / 4 is less than H / 2 and ml is positive, the weighted average y must be less than H / 2, meaning the centre of gravity moves downward from its original position.

Verification / Alternative check:
Consider extreme cases. If the mass of the liquid is very small compared to the cylinder, the position of the combined centre will be only slightly lower than H / 2, but still lower. If the mass of the liquid is very large compared to the cylinder, the combined centre will come close to H / 4, the position of the liquid centre. In every possible case with some liquid present, the combined centre must lie between H / 4 and H / 2 and therefore below H / 2. This confirms that the centre of gravity always shifts downward when a liquid is added to the lower part of the cylinder.

Why Other Options Are Wrong:
Shifted upwards is impossible, because all added mass lies below the original centre of gravity.
Shifted neither upward nor downward can occur only if no new mass is added or it is added exactly at the original centre height, which is not the case.
Shifted upward or downward depending on density is incorrect because any finite liquid mass at a lower level pulls the centre down, regardless of exact density.
Moves to the very bottom is wrong; the centre of gravity will approach the lower region but cannot lie at the extreme bottom while some mass remains above.

Common Pitfalls:
Students sometimes think that if the cylinder itself is heavy, the centre of gravity cannot move, which is not correct. Adding even a small mass in a lower position will still pull the centre slightly downward. Others may incorrectly assume the location depends on density direction in a more complicated way. The key idea is that any additional mass below the original centre will lower the combined centre of gravity, improving stability.

Final Answer:
When the cylinder is half filled with liquid, the combined centre of gravity is shifted downwards compared to the empty cylinder.