Question

The gravitational force exerted on a solid object is $\mathbf{\text{5.00N}}$. When the object is suspended from a spring scale and submerged in water, the scale reads$\mathbf{3}\mathbf{.}\mathbf{50}\mathbf{\text{N}}$ (Fig. P14.26). Find the density of the object.

Short answer

The density of the object is${\rho }_0=3.33\times {10}^3\raisebox{1ex}{$\text{kg}$}\!\left/ \!\raisebox{-1ex}{${\text{m}}^{\text{3}}$}\right.$.

Step-by-step solution

Given Data:

The gravitationalforceonobject,$F=mg=5\text{N}$

The mass of object,$M=0.5\text{\hspace{0.17em}kg}$

Reading when it submerged in water,$F\text{'}=3.5\text{\hspace{0.17em}N}$

The loss in weight,$W=5-3.5=1.5\text{\hspace{0.17em}N}$

A concept:

When an object is partially or completely submerged in a fluid, the fluid exerts an upward force on the object called buoyancy. According to Archimedes' principle, the magnitude of the buoyant force is equal to the weight of the fluid displaced by the object.

$B={\rho }_{Fluid}g{V}_{disp}$

Where, role="math" localid="1663743372476" $V_{disp}$is the volume of fluid, $g$is the acceleration due to gravity $\text{10}\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{${\text{s}}^{\text{2}}$}\right.$, and role="math" localid="1663743412511" ${\rho }_{Fluid}$is the density of the water having a value $1000\raisebox{1ex}{$\text{kg}$}\!\left/ \!\raisebox{-1ex}{${\text{m}}^{\text{3}}$}\right.$.

Find when the object is suspended from a spring scale and submerged in water:

The loss in weight is due to buoyancy of water, thus loss in weight of object is equal to the weight of displaced water.

$W=1.5\text{\hspace{0.17em}N}$

Also the weight of displaced water is,

$$\begin{gathered} W={\rho }_w\times V\times g \\ 1.5=1000\times V\times 10 \\ V=\frac{1.5}{{10}^4}{\text{m}}^{\text{3}} \end{gathered}$$

Now, for object let its density be ${\rho }_0$. Therefore,

$\begin{array}{rcl}{\rho }_0& =& \frac{m}{V}\\ & =& \frac{0.5\times {10}^4}{1.5}\\ & =& 3.33\times {10}^3\raisebox{1ex}{$\text{kg}$}\!\left/ \!\raisebox{-1ex}{${\text{m}}^{\text{3}}$}\right.\end{array}$

Hence, the density of the object is$3.33\times {10}^3\raisebox{1ex}{$\text{kg}$}\!\left/ \!\raisebox{-1ex}{${\text{m}}^{\text{3}}$}\right.$.