Question

A plastic ball is held at the bottom of a bucket of water and then released. As it is released, the bucket is dropped over the edge of a cliff. What happens?

Short answer

Short Answer: When the bucket and the plastic ball are dropped off a cliff, the ball will remain submerged in the water and "float" inside the bucket while falling towards the ground. This occurs because the gravitational force acting on the ball, water, and bucket is the same, causing them to fall at the same rate, and there is no buoyant force acting on the ball during free fall.

Step-by-step solution

Analyze the initial situation

Initially, the plastic ball is submerged in water and held at the bottom of the bucket. Two primary forces are acting on the ball: the buoyant force (upward direction) and the gravitational force (downward direction). The buoyant force is given by: F_b = ρ_water * V_ball * g where ρ_water is the density of water, V_ball is the volume of the ball, and g is the acceleration due to gravity. On the other hand, the gravitational force acting on the ball, also known as weight, is given by: F_g = m_ball * g where m_ball is the mass of the plastic ball. Since the ball is held in place, the net force acting on it is zero. But once it is released, the ball will experience a net buoyant force that will push it upward since F_b > F_g (as it is a plastic ball).

Analyze the situation when the bucket is dropped

When the bucket is dropped off the cliff, both the bucket and the ball fall freely under the influence of gravity. They experience the same acceleration due to gravity (g). Since there is no additional force acting on the water inside the falling bucket in the vertical direction, there won't be any upward buoyant force on the ball.

Determine the behavior of the ball

As there is no buoyant force acting on the ball while the bucket is falling, the ball will also be in free fall along with the water and the bucket. The gravitational force acting on the ball will be the same as the gravitational force acting on the water and the bucket. Therefore, the ball, water, and the bucket will fall at the same rate, and thus the plastic ball will remain submerged and appear to float inside the bucket as they all fall together. So, when the bucket containing the ball and the water is dropped, the ball will remain submerged in the water and they will "float" inside the bucket while falling towards the ground.

Key concepts

Buoyant Force

Buoyant force is a concept in physics that explains why objects float or sink in a fluid, such as water. This force acts in the upward direction against the weight of the object. It's caused by the pressure difference between the top and bottom of the submerged object. The greater the volume of the object submerged, the greater the buoyant force.
The principle behind buoyancy is known as Archimedes' Principle, which states that any object wholly or partially immersed in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. The formula for buoyant force is:

• \( F_b = \rho_{\text{water}} \cdot V_{\text{ball}} \cdot g \)

where \( \rho_{\text{water}} \) is the density of water, \( V_{\text{ball}} \) is the volume of the ball, and \( g \) is the acceleration due to gravity.
In this exercise, initially the plastic ball experiences a buoyant force because it is submerged in water. This force attempts to push the ball upwards, opposing the gravitational pull.

Gravitational Force

Gravitational force, or weight, is the force of attraction between two masses. In simple terms, it's what keeps us on the ground on Earth and is caused by the mass of Earth pulling objects downward. The formula for this force in physics is:

• \( F_g = m_{\text{ball}} \cdot g \)

where \( m_{\text{ball}} \) is the mass of the ball and \( g \) is the acceleration due to gravity, typically approximated as \( 9.81 \, \text{m/s}^2 \) on Earth.
In the context of our exercise, the gravitational force acts downwards on both the bucket and the ball. Initially, while the ball is at the bottom of the bucket, the gravitational force is balanced by the buoyant force. However, once released, and as the bucket is allowed to fall, gravity acts uniformly on everything, contributing to a free-fall scenario.

Free Fall

Free fall is a unique type of motion where gravity is the only force acting on an object. This situation is observed when objects are dropped from a height with no significant air resistance, allowing them to accelerate downwards at \( 9.81 \, \text{m/s}^2 \).
When the bucket containing water and the plastic ball is dropped over the cliff, everything inside the bucket experiences free fall together. As all parts are falling under the same gravitational acceleration, the relative motion between them is minimal.
_{Key Points About Free Fall:}

• All objects fall at the same rate regardless of their mass in the absence of air resistance.
• During free fall, the only force acting on the objects is gravity.

In the exercise, free fall means that the buoyant force ceases to act on the ball, as it is no longer counteracting gravity in the falling frame of reference. The ball "floats" inside the liquid as the whole system accelerates downwards uniformly.

Physics Education

Physics education aims to make foundational concepts, such as forces and motion, accessible and comprehensible to learners. It teaches students how to apply theoretical principles to real-world scenarios, enhancing problem-solving skills.
Understanding dynamics problems like the one in this exercise helps students grasp the interplay of forces acting within a system. By breaking down forces into buoyant and gravitational, and considering the conditions like free fall, students can predict object behavior.

• Exposing students to a variety of physics problems builds intuition about natural phenomena.
• Enhanced problem-solving skills from physics can apply to a broad range of careers.
• Interactive learning through experiments makes abstract concepts like buoyancy and free fall tangible.

This exercise, involving buoyancy and free fall, illustrates the balance and unison of forces. By understanding this balance, students can better grasp how forces affect motion in various contexts, fostering a deeper appreciation for physics.