To analyze the situation with the cylindrical container, the liquid, and the floating ball, we need to consider the principles of buoyancy and how they interact with the forces acting on the system. Let’s break down the scenario step by step.

Understanding the Forces at Play

When the ball is submerged in the liquid, two main forces come into play: the buoyant force acting on the ball and the weight of the ball itself. The buoyant force can be described by Archimedes' principle, which states that the upward buoyant force exerted on a submerged object is equal to the weight of the fluid that the object displaces.

Calculating the Buoyant Force

The buoyant force (F_b) can be calculated using the formula:

• F_b = ρ × g × V

Where:

• ρ is the density of the liquid.
• g is the acceleration due to gravity.
• V is the volume of the ball.

Since the ball is floating, it displaces a volume of liquid equal to its own volume, V. Therefore, the buoyant force acting on the ball is equal to the weight of the liquid displaced by the ball.

Weight of the Ball

The weight of the ball (W_b) is given by:

• W_b = m × g

Where m is the mass of the ball. Since the ball is being held underwater, we need to consider how this affects the scale reading.

Impact on the Scale Reading

The scale measures the total weight acting on it, which includes the weight of the liquid and the weight of the ball. When the ball is submerged, the scale experiences an additional force due to the buoyant force acting on the ball. However, since the ball is being held down, the buoyant force does not cause the ball to rise, but it does contribute to the overall weight measured by the scale.

Net Weight on the Scale

The total weight (W_total) on the scale can be expressed as:

• W_total = W + W_b

Here, W is the initial weight of the liquid in the container. When the ball is submerged, the scale reading increases by the weight of the ball since the buoyant force is countered by the force applied to hold the ball down. Thus, the scale reads:

• W_total = W + m × g

Conclusion

In summary, when the ball is submerged and held beneath the surface, the scale reading increases by the weight of the ball. This is because the buoyant force acting on the ball is countered by the force you apply to keep it submerged, resulting in an increase in the total weight measured by the scale. Understanding these principles allows us to predict how the system behaves under different conditions, such as changing the volume or mass of the ball or the density of the liquid.