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Chapter 10: Problem 47

An aluminium sphere is dipped into water. Which of the following is true? (A) Buoyancy will be less in water at \(0^{\circ} \mathrm{C}\) than that in water at \(4^{\circ} \mathrm{C}\) (B) Buoyancy will be more in water at \(0^{\circ} \mathrm{C}\) than that in water at \(4^{\circ} \mathrm{C}\) (C) Buoyancy in water at \(0^{\circ} \mathrm{C}\) will be same as that in water at \(4^{\circ} \mathrm{C}\) (D) Buoyancy may be more or less in water at \(4^{\circ} \mathrm{C}\) depending on the radius of the sphere

Short Answer

Expert verified
The correct answer is (A): The buoyancy will be less in water at \(0^{\circ} \mathrm{C}\) than that in water at \(4^{\circ} \mathrm{C}\). This is because water is denser at \(4^{\circ} \mathrm{C}\), leading to a greater buoyant force at this temperature.

Step by step solution

01

Understand the buoyant force

The buoyant force on an object submerged in a fluid depends on four things: the volume of the object, the volume of fluid displaced, the density of the fluid, and the gravitational force. It's given by the formula \(F = V \times d \times g\), where \(F\) is the buoyant force, \(V\) is the volume of the fluid displaced (which also is the volume of the object if completely immersed), \(d\) is the density of the fluid, and \(g\) is the acceleration due to gravity.
02

Consider the effect of temperature on density

The density of water is maximum at \(4^{\circ} \mathrm{C}\), and decreases both above and below this temperature. So, water is denser at \(4^{\circ} \mathrm{C}\) than at \(0^{\circ} \mathrm{C}\). This means that the same volume of water would weigh more at \(4^{\circ} \mathrm{C}\) than at \(0^{\circ} \mathrm{C}\). Consequently, the buoyant force (which is equal to the weight of the displaced fluid) will be greater at \(4^{\circ} \mathrm{C}\) than at \(0^{\circ} \mathrm{C}\).
03

Compare the buoyant forces at different temperatures

Since both the volume of the sphere (and thus the volume of water displaced) and the gravitational force remain constant while dipping the sphere in water at \(0^{\circ} \mathrm{C}\) and \(4^{\circ} \mathrm{C}\), the difference in buoyancy in these two cases comes solely from the different densities of the water. From the reasoning in Step 2, it can be deduced that the buoyant force will be higher in water at \(4^{\circ} \mathrm{C}\) than at \(0^{\circ} \mathrm{C}\).

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