Problem 47 - 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

(c) \(w_{1}

Step 1: Understand the Relationship between Weight and Density

When the metal ball is immersed in alcohol, it experiences a buoyant force due to the displacement of the liquid, which affects its weight. The weight of the ball in the liquid is given by the actual weight of the ball minus the buoyant force.

Step 2: Examine the Effect of Temperature on Buoyant Force

As temperature increases, the density of the alcohol changes because of its higher coefficient of cubical expansion compared to the metal. Therefore, as temperature increases, the alcohol becomes less dense, reducing the buoyant force on the metal ball.

Step 3: Analyze the Effects of Lesser Dense Alcohol on Weight

As the alcohol density decreases with temperature increase, the buoyant force exerted on the metal decreases. This implies that the weight of the metal ball should increase as the temperature goes up, assuming the weight is measured as apparent weight in the fluid.

Step 4: Conclude the Relationship between Weights at Different Temperatures

Given the above analysis, the metal ball will weigh less when initially submerged at lower temperatures compared to at higher temperatures. Thus, the apparent weight at 0°C (\(w_1\)) will be less than the apparent weight at 59°C (\(w_2\)). Therefore, \(w_1 < w_2\).

Key Concepts

Cubical Expansion

When materials are heated, they expand. This expansion can occur in three dimensions, a phenomenon known as cubical expansion. The cubical expansion of a substance is quantified by its coefficient of cubical expansion, which tells us how much its volume will change with temperature. In the context of the metal ball immersed in alcohol, both the ball and the alcohol undergo cubical expansion as their temperature increases. However, the alcohol has a higher coefficient of cubical expansion than the metal ball. This means that for the same temperature increase, the volume change in alcohol is greater compared to that of the metal ball. This difference plays a crucial role in the buoyant force and thus affects the apparent weight of the metal ball within the alcohol.

Density

Density is a property of matter that expresses its mass per unit volume, given as \( \frac{mass}{volume} \). It helps determine how substances behave when placed in a fluid. In the situation of the metal ball and alcohol, density influences how much buoyant force acts on the ball. - When alcohol is denser, it can exert a greater buoyant force.- When alcohol is less dense due to temperature changes, the buoyant force decreases.This relationship between density and the weight of an object in a fluid is essential in understanding buoyancy. In this scenario, the higher initial density of alcohol at a lower temperature results in a strong buoyant force, making the ball appear lighter.

Temperature Effect on Density

The relationship between temperature and density is pivotal in understanding changes in buoyant forces. Generally, as the temperature of a substance increases, its density decreases. This is because heating a fluid gives its molecules more kinetic energy, causing them to spread apart, and thereby decreasing density. In our exercise, the metal ball is suspended in alcohol, and as the temperature of alcohol increases from \(0^{\circ} \mathrm{C}\) to \(59^{\circ} \mathrm{C}\), the alcohol density decreases due to its high coefficient of cubical expansion. - At higher temperatures, the alcohol is less dense.- This decrease in density reduces the buoyant force on the metal ball.The change is significant as it alters the apparent weight of the metal ball, making it appear heavier at higher temperatures compared to lower ones. By analyzing these effects, we can conclude that the apparent weight of the ball increases with temperature rise.

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Problem 46

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Key Concepts

Cubical Expansion

When materials are heated, they expand. This expansion can occur in three dimensions, a phenomenon known as cubical expansion. The cubical expansion of a substance is quantified by its coefficient of cubical expansion, which tells us how much its volume will change with temperature. In the context of the metal ball immersed in alcohol, both the ball and the alcohol undergo cubical expansion as their temperature increases. However, the alcohol has a higher coefficient of cubical expansion than the metal ball. This means that for the same temperature increase, the volume change in alcohol is greater compared to that of the metal ball. This difference plays a crucial role in the buoyant force and thus affects the apparent weight of the metal ball within the alcohol.

Density

Density is a property of matter that expresses its mass per unit volume, given as \( \frac{mass}{volume} \). It helps determine how substances behave when placed in a fluid. In the situation of the metal ball and alcohol, density influences how much buoyant force acts on the ball. - When alcohol is denser, it can exert a greater buoyant force.- When alcohol is less dense due to temperature changes, the buoyant force decreases.This relationship between density and the weight of an object in a fluid is essential in understanding buoyancy. In this scenario, the higher initial density of alcohol at a lower temperature results in a strong buoyant force, making the ball appear lighter.

Temperature Effect on Density

The relationship between temperature and density is pivotal in understanding changes in buoyant forces. Generally, as the temperature of a substance increases, its density decreases. This is because heating a fluid gives its molecules more kinetic energy, causing them to spread apart, and thereby decreasing density. In our exercise, the metal ball is suspended in alcohol, and as the temperature of alcohol increases from \(0^{\circ} \mathrm{C}\) to \(59^{\circ} \mathrm{C}\), the alcohol density decreases due to its high coefficient of cubical expansion. - At higher temperatures, the alcohol is less dense.- This decrease in density reduces the buoyant force on the metal ball.The change is significant as it alters the apparent weight of the metal ball, making it appear heavier at higher temperatures compared to lower ones. By analyzing these effects, we can conclude that the apparent weight of the ball increases with temperature rise.