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Physics: Motion and Mechanics - Density, Flotation, and Sinking

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Density is defined as the mass per unit volume of a substance. It is a characteristic property of a material and is calculated using the formula ρ=mV\rho = \frac{m}{V}.

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The SI unit for density is kilograms per cubic meter (kg/m3kg/m^3), though grams per cubic centimeter (g/cm3g/cm^3) is frequently used. Note that 1 g/cm3=1000 kg/m31\ g/cm^3 = 1000\ kg/m^3.

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To measure the density of an irregular solid, the displacement method is used: the volume of the solid is equal to the volume of the liquid it displaces (often measured using a displacement/Eureka can).

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Upthrust (or Buoyancy) is the upward force exerted by a fluid on any object immersed in it. It acts in the opposite direction to gravity.

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Archimedes' Principle states that the upthrust acting on an object is equal to the weight of the fluid displaced by the object.

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The Law of Flotation states that a floating object displaces a weight of fluid equal to its own weight. This occurs when the object's density is less than or equal to the fluid's density.

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An object will sink if its density is greater than the density of the fluid (ρobject>ρfluid\rho_{object} > \rho_{fluid}), and it will float if its density is less than the fluid (ρobject<ρfluid\rho_{object} < \rho_{fluid}).

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Relative Density (Specific Gravity) is the ratio of the density of a substance to the density of water at 4∘C4^\circ C. It has no units.

📐Formulae

ρ=mV\rho = \frac{m}{V}

m=ρ×Vm = \rho \times V

V=mρV = \frac{m}{\rho}

Upthrust=Weight of displaced fluid\text{Upthrust} = \text{Weight of displaced fluid}

Relative Density=Density of substanceDensity of water\text{Relative Density} = \frac{\text{Density of substance}}{\text{Density of water}}

💡Examples

Problem 1:

A solid rectangular block of metal has a mass of 400 g400\ g. Its dimensions are 5 cm×4 cm×2 cm5\ cm \times 4\ cm \times 2\ cm. Calculate the density of the metal in g/cm3g/cm^3 and determine if it will float in water (density of water =1 g/cm3= 1\ g/cm^3).

Solution:

First, calculate the volume of the block: V=5 cm×4 cm×2 cm=40 cm3V = 5\ cm \times 4\ cm \times 2\ cm = 40\ cm^3 Next, use the density formula: ρ=mV=400 g40 cm3=10 g/cm3\rho = \frac{m}{V} = \frac{400\ g}{40\ cm^3} = 10\ g/cm^3

Explanation:

Since the density of the metal (10 g/cm310\ g/cm^3) is greater than the density of water (1 g/cm31\ g/cm^3), the block will sink.

Problem 2:

A piece of wood with a volume of 0.05 m30.05\ m^3 floats in a lake. If the density of the wood is 600 kg/m3600\ kg/m^3, what is the mass of the wood?

Solution:

Using the rearranged density formula: m=ρ×Vm = \rho \times V m=600 kg/m3×0.05 m3=30 kgm = 600\ kg/m^3 \times 0.05\ m^3 = 30\ kg

Explanation:

The mass is found by multiplying the density of the material by the total volume of the object.

Problem 3:

An object weighs 15 N15\ N in air. When completely submerged in water, it weighs 12 N12\ N. Calculate the upthrust acting on the object.

Solution:

15 N−12 N3 N\begin{array}{r} 15\ N \\ -12\ N \\ \hline 3\ N \end{array} Upthrust=Weight in air−Apparent weight in liquid=3 N\text{Upthrust} = \text{Weight in air} - \text{Apparent weight in liquid} = 3\ N

Explanation:

The loss in weight of the object when submerged is exactly equal to the upthrust exerted by the water, according to Archimedes' Principle.