Question

You fill a tall glass with ice and then add water to the level of the glass's rim, so some fraction of the ice floats above the rim. When the ice melts, what happens to the water level? (Neglect evaporation, and assume that the ice and water remain at \(0^{\circ} \mathrm{C}\) during the melting process.) a) The water overflows the rim. b) The water level drops below the rim. c) 'The water level stays at the rim. d) It depends on the difference in density between water and ice.

Short answer

a) The water level increases. b) The water level decreases. c) The water level stays at the rim. d) Not enough information. Answer: c) The water level stays at the rim.

Step-by-step solution

Understand the given information

In the problem, a tall glass is filled with ice and water. The water level reaches the glass's rim, and some of the ice floats above the rim. We need to find out what happens to the water level as the ice melts, without considering evaporation or temperature change.

Apply Archimedes' principle

According to Archimedes' principle, 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, the weight of the ice (including the part that floats above the rim) displaces an equal weight of water, which means the melted ice will exactly replace the water initially displaced by the solid ice, maintaining the water level. Mathematically, the weight of the ice is given by: \(W_i = m_i \times g\) Where \(W_i\) is the weight of the ice, \(m_i\) is the mass of the ice, and \(g\) denotes the gravitational acceleration. Similarly, the weight of the water initially displaced by the ice is: \(W_w = m_w \times g\) Where \(W_w\) is the weight of the displaced water, and \(m_w\) is the mass of the displaced water. According to Archimedes' principle, \(W_i = W_w\).

Compare the melting of ice and the weight of water

When the ice melts, it turns into an equal mass of water. Since the mass of water obtained after melting and the mass of water initially displaced are equal, the overall water level remains unchanged. Mathematically, we can show this by: \(M_{melted\_ice} = M_{displaced\_water}\) Where \(M_{melted\_ice}\) and \(M_{displaced\_water}\) denote the mass of the water from melted ice and the mass of the initially displaced water, respectively.

Find the answer

Considering the above analysis, we can confidently conclude that when the ice melts, the water level stays at the rim. Therefore, the correct answer is: c) The water level stays at the rim.

Key concepts

Buoyant Force

To understand why objects float or sink in water, we must grasp the concept of buoyant force. When an object is placed in a fluid, it displaces a certain volume of that fluid. According to Archimedes' principle, the fluid exerts an upward force on the object equal to the weight of the fluid displaced. This upward force is known as the buoyant force. It's crucial when analyzing why ice floats on water and what happens during the melting process.

If the weight of the object is less than the buoyant force, the object will float. If it's greater, the object will sink. For instance, ice floats on water because the buoyant force is greater than the weight of the ice due to the solid's lower density compared to that of water.

Density of Water and Ice

The unusual relationship between the densities of water and ice plays a pivotal role in the behavior of melting ice in water. Density is defined as mass per unit volume. Water reaches its maximum density at about 4 degrees Celsius. Below this temperature, water starts to expand, and thus its density decreases. This is why ice, which forms when water is below 0 degrees Celsius, has a lower density than liquid water.

Due to this density difference, ice floats on water. Each ice cube displaces a volume of water that weighs the same as the ice. Therefore, when the ice melts, it converts into a volume of water equal to the volume it previously displaced, thereby keeping the water level constant.

Melting Ice Experiment

The melting ice experiment, such as the one described in the exercise, is a classic demonstration of Archimedes' principle and density concepts. The setup involves ice floating in water, which inherently challenges us to predict what will happen when the ice melts. As the ice changes from solid to liquid form, it replaces the volume of water it had displaced when it was frozen. There’s no increase in the water level because the mass of water produced by the melting ice is exactly equal to the mass of the ice that was floating. This outcome elegantly underscores the Archimedes' principle, showing how the buoyant force and densities are imperatively connected.

Gravitational Acceleration

Gravitational acceleration, denoted by the symbol 'g', is the acceleration that is imparted to objects due to the force of gravity exerted by a massive body like the Earth. It's a fundamental force that affects every object with mass. At the Earth's surface, 'g' is approximately 9.81 meters per second squared. This value is crucial when calculating the weight of objects, which is the product of mass (m) and gravitational acceleration (g).

In our ice and water problem, 'g' helped us understand that the weight of the ice before and after melting remains constant. This constancy ensures that the water level will not change as the ice converts to liquid form because the force applied by the water--the buoyant force--equally counters the gravitational force, maintaining equilibrium and stabilizing the water level in the glass.