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Question

A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces 89.0 cm³ of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes' principle.)

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From  the definition of the average density\textbf{ the definition of the average density} we know that :Density The average density of a substance or object is defined as its mass per unit volume

ρ=mV\rho = \dfrac{m}{V}

, Archimedes’ Principle\textbf{Archimedes’ Principle} The buoyant force on an object equals the weight of the fluid it displaces.

FB=wflF_{B} = w_{fl}

Where:

  • FBF_{B} is the buoyant force .
  • wflw_\text{fl} is the weight of the displaced water .
  • ρrock\rho_\text{rock} is the average density of the rock .
  • ρfl\rho_\text{fl} is the average density of the water.
  • mrockm_\text{rock} is the mass of the rock.
  • mflm_\text{fl} is the mass of the water.
  • VrockV_\text{rock} is the volume of the rock

Givens\textbf{Givens}: ρfl=1g/cm3\rho_\text{fl} = 1 \mathrm{g/cm^3} , V=89cm3V =89 \mathrm{cm^3} .

Plugging\textbf{Plugging} known information to get :

FB=wflmrockg=mflgmrock=ρflVflρrock=ρflVflVrock=1g/cm3×89cm3Vrock=89\begin{align*} F_\text{B}& = w_\text{fl} \\ m_\text{rock} g &= m_\text{fl} g\\ m_\text{rock} &= \rho_\text{fl} V_\text{fl} \\ \rho_\text{rock} &= \dfrac{\rho_\text{fl} V_\text{fl}}{ V_\text{rock}} \\ &= \dfrac{1 \mathrm{g/cm^3} \times 89 \mathrm{cm^3} }{ V_\text{rock}} \\ &= 89 \end{align*}

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