Question

A beaker of water is sitting on a scale. A steel ball hanging from a string is lowered into the water until the ball is completely submerged but is not touching the beaker. The weight registered by the scale will a) increase. b) decrease c) stay the same.

Short answer

Answer: The weight registered by the scale will increase.

Step-by-step solution

Understanding Archimedes' principle

Archimedes' principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In this case, when the steel ball is submerged into the water, it displaces a certain volume of water, which will experience an upward buoyant force equal to its weight.

Analyzing the forces acting on the beaker with water and the steel ball

When the steel ball is not submerged in the water, the only force acting on the beaker with water is its weight (downward), and the scale registers this weight. When the steel ball is submerged in the water, there is a buoyant force acting upward on the ball, which is equal to the weight of the water displaced by the ball. This upward force reduces the net downward force acting on the ball.

Determining the effect of the submerged ball on the weight registered by the scale

As the buoyant force acts on the ball, the ball, in turn, exerts a force on the water (which is equal in magnitude but in the opposite direction) due to Newton's third law. The water, therefore, exerts an additional downward force on the beaker. This additional force increases the total downward force acting on the beaker, which means that the weight registered by the scale will increase. Therefore, the answer is (a) increase.

Key concepts

Buoyant Force

When an object is submerged in a fluid, it experiences an upward force known as the buoyant force. This force is crucial to understanding why objects float or sink. Essentially, Archimedes' principle tells us that the buoyant force on a submerged object is equal to the weight of the fluid it displaces. Let's imagine we place a steel ball into water. The water makes way for the ball, and the volume of water displaced has a certain weight. According to the principle, the ball now experiences an upward force equal to this weight. Consequently, if the weight of the discharged water is more than the weight of the ball, the ball floats. Otherwise, it sinks but still gets support from the buoyant force, which is less than its own weight.

In our exercise, as the ball is submerged, the upward buoyant force exerted by the water opposes some of the ball's weight. This might seem as if it would decrease the scale reading, but we need to consider the system as a whole, including the water and the beaker.

Fluid Displacement

To fully comprehend the concept of fluid displacement, let's zoom in on what happens when we immerse an object into a fluid like water. Displacement is when an object pushes the fluid out of the way, taking up space that was previously occupied by the fluid. Think of a full bathtub – when you step in, water spills out because your body has displaced it. This displaced fluid has to go somewhere, and in our text-book exercise, the water level in the beaker rises as the steel ball is submerged.

The key to understanding this concept is recognizing that displacement connects to volume – the amount of space something takes up – and not the weight of the displacing object. Thus, the assistive upward force caused by displacement (buoyant force) is only related to the volume of the fluid moved and its weight, not the submerged object's weight or volume. In our exercise case, the volume of water displaced by the steel ball has an immediate and required relevance to the observed change on the scale.

Newton's Third Law of Motion

Newton's third law of motion states that for every action, there is an equal and opposite reaction. This aspect of physics is integral to understanding the interactions between objects and their environments. In the context of the exercise, when the steel ball is submerged, it's not just the ball that's affected by the water; the water is also affected by the ball. The force exerted by the ball downwards (due to gravity) is met by an equal and opposite force upwards from the water. This relationship exists in every interaction where forces are at play.

When the steel ball, dangling from a string, is lowered into the water, it exerts a downward force on the water. Because of Newton's third law, the water pushes back with an equal force in the opposite direction. This force being exerted on the water is transferred to the beaker, and hence, to the scale, resulting in an increased reading on the scale. It is crucial to recognize that forces are always in pairs and affect all objects involved in the interaction, not just the most obvious one - in our case, not just the ball, but also the water and the beaker.