Problem 46 - Master Resource Book in JEE Main Physics

Question

A metal ball immersed in alcohol weighs \(w_{1}\) at \(0^{\circ} \mathrm{C} \quad \overline{5}\) and \(w_{2}\) at \(59^{\circ} \mathrm{C}\). The coefficient of cubical expansion of the metal is less than that of alcohol. Assuming that the density of metal is large compared to that of alcohol, it can be shown that (a) \(w_{1}>w_{2}\) (b) \(w_{1}=w_{2}\) (c) \(w_{1}

Step-by-Step Solution

Answer

(a) \( w_1 > w_2 \)

Step 1: Understanding the Problem

The exercise involves a metal ball immersed in alcohol, leading to changes in weight with temperature changes. The key factors are the cubical expansion coefficients of the metal and alcohol, and the density of the metal relative to alcohol.

Step 2: Identify Cubical Expansion Impact

As temperature increases, both the metal ball and the alcohol expand. The coefficient of cubical expansion is higher for alcohol than for the metal, causing the alcohol volume to increase more than the metal's volume.

Step 3: Apply the Buoyancy Principle

Buoyancy depends on the volume of displaced liquid. As the alcohol expands more than the metal, a larger volume of alcohol is displaced at higher temperatures, reducing the net weight of the metal ball.

Step 4: Determine the Weight Change

The weight of the ball at 59°C, denoted by \( w_2 \), will be less than the weight at 0°C, denoted by \( w_1 \), because more alcohol is displaced at the higher temperature due to greater expansion.

Key Concepts

Cubical Expansion

When we discuss cubical expansion, we refer to the way a material's volume changes as its temperature changes. Imagine each side of a cube getting slightly longer as it warms. The volume increase for different substances varies depending on their coefficient of cubical expansion.

For example, alcohol has a higher cubical expansion coefficient than metal. This means alcohol expands more for the same temperature increase compared to metal. In our exercise, as the metal ball and alcohol are heated, the alcohol expands significantly more than the metal. Understanding cubical expansion helps explain why the metal ball appears lighter at a higher temperature when immersed in alcohol.

Key points worth noting include:

• Cubical expansion measures volume change with temperature.
• Substances expand at different rates.
• Higher coefficients mean greater expansion.

Density

Density is a fundamental property of matter: a measure of mass per unit volume. It's expressed in units like kg/m³. The density of a substance can tell us how tightly packed its molecules are.

In our scenario, the density of the metal ball is considerably higher than that of alcohol. This means the metal ball's mass per volume is greater, leading to significant implications in buoyancy and weight changes with temperature. As the liquid around expands, it's crucial to note that density variations play a key role in determining the ball's effective weight in the liquid.

Here's what to remember:

• Higher density = more mass per volume.
• Metal generally denser than alcohol, affecting buoyancy.
• Density constant for solids but can change for liquids with temperature.

Thermal Expansion

Thermal expansion refers to how substances change dimensions with temperature fluctuations. Metals and liquids like alcohol expand when heated. The linear expansion in solids is less noticeable than the cubical expansion in liquids.

In diverse materials, the extent of thermal expansion varies. For our exercise, the crucial point is realizing differential thermal expansion. Alcohol’s greater expansion than metal affects buoyancy, leading to different readings at varying temperatures.

Key points are:

• Thermal expansion causes size changes with temperature.
• Liquids generally expand more than solids with heat.
• Differential expansion affects buoyancy and weight.

Archimedes' Principle

Archimedes' principle is the backbone of understanding buoyancy. It states that a body submerged in a fluid experiences an upward force equal to the weight of the fluid displaced by the body.

Relating this to our problem, as the temperature rises, alcohol around the metal ball expands faster than the ball itself. The additional volume displaces more fluid, thus increasing buoyancy and seeming lighter weight observations for the ball at higher temperatures.

Remember these essentials for Archimedes' principle:

• Buoyant force = weight of displaced fluid.
• Buoyancy affects how heavy or light things feel in fluid.
• Greater fluid expansion can lead to more fluid displacement.