Question

In the figure, a cube of edge length L = 0.599 m and mass 970 kg is suspended by a rope in an open tank of liquid of density 1.05E+3 kg/m3. Find (a) t

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asked 2021-04-19
In the figure, a cube of edge length L = 0.599 m and mass 970 kg is suspended by a rope in an open tank of liquid of density 1.05E+3 kg/m3. Find (a) the magnitude of the total downward force on the top of the cube from the liquid and the atmosphere, assuming atmospheric pressure is 1.00 atm, (b) the magnitude of the total upward force on the bottom of the cube, and (c) the tension in the rope. (d) Calculate the magnitude of the buoyant force on the cube using Archimede's principle.
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Answers (1)

2021-04-21
a) \(\displaystyle{P}_{{{a}{t}{m}}}={1}\ {a}{t}{m}={101325}\ \frac{{N}}{{m}^{{2}}}\)
Pressure due to water \(\displaystyle=\rho{g}{h}={\left({1.05}\cdot{10}^{{3}}\right)}{\left({9.81}\right)}{\left({\frac{{{0.599}}}{{{2}}}}\right)}={3085}\frac{{m}}{{m}^{{2}}}\)
Net pressure \(\displaystyle={101325}+{3085}={104410}\ \frac{{N}}{{m}^{{2}}}\)
Cross sectional area of the cube \(\displaystyle={L}^{{2}}={0.36}{m}^{{2}}\)
Net force \(\displaystyle={P}\cdot{A}={37587.6}{N}\)
b) \(\displaystyle{P}_{{{a}{t}{m}}}={1}\ {a}{t}{m}={101325}\frac{{N}}{{m}^{{2}}}\)
Pressure due to water \(\displaystyle=\rho{g}{h}={\left({1.05}\cdot{10}^{{3}}\right)}{\left({9.81}\right)}{\left({3}\cdot{\frac{{{0.599}}}{{{2}}}}\right)}={9255}\ \frac{{N}}{{m}^{{2}}}\)
Net pressure =101325+9255=110580 \(\displaystyle\frac{{N}}{{m}^{{2}}}\)
Cross sectional area of the cube \(\displaystyle={L}^{{2}}={0.36}{m}^{{2}}\)
Net force \(\displaystyle={P}\cdot{A}={39808.8}{N}\)
c) Weight of the cube \(\displaystyle={m}{g}={970}{c}{d}{\quad\text{or}\quad}{9.81}={9515.7}\ {N}\)
\(\displaystyle{T}={37587.6}+{9515.7}={4703.3}{N}\)
d) Buoyant force \(\displaystyle={V}\rho{g}\)
\(\displaystyle\Rightarrow{F}_{{B}}={\left({0.599}^{{3}}\right)}{\left({1.05}\cdot{10}^{{3}}\right)}{\left({9.81}\right)}={2213.80}\ {N}\)
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