A cube 80 mm on a side is made of a rigid foam material and floats in water with 60 mm of the cube below the surface. Calculate the magnitude and direction of the force required to hold it completely submerged in glycerin, which has a specific gravity of 1.26.

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Question

A cube 80 mm on a side is made of a rigid foam material and floats in water with 60 mm of the cube below the surface. Calculate the magnitude and direction of the force required to hold it completely submerged in glycerin, which has a specific gravity of 1.26.

Complete the solution before looking at the next panel.

Accepted Answer

First calculate the weight of the cube, then the force required to hold the cube submerged in glycerin. Use the free-body diagrams in Fig. 5.7: (a) a cube floating on water and (b) a cube submerged in glycerin.

From Fig. 5.7(a), we have

[latex]\sum{F_v}= 0 [/latex]

[latex]F_b - W =0[/latex]

[latex]W=F_b=\gamma_f V_d[/latex]

[latex]V_d =\left(80mm ight) \left(80mm ight) \left(60mm ight) =384 imes 10^3 mm^3[/latex]

(submerged volume of cube)

[latex]W = \left(\frac{9.81 imes 10^3 N}{m^3} ight) \left(384 imes 10^3 mm^3 ight) \left(\frac{1m^3}{\left(10^3 mm ight)^3 } ight) [/latex]

= 3.77 N

From Fig. 5.7(b), we have

[latex]\sum{F_v}= 0 [/latex]

[latex]F_b - F_e -W=0[/latex]

[latex] F_e= F_b - W =\gamma_f V_d-3.77N[/latex]

[latex]V_d = \left(80mm ight)^3 =512 imes 10^3 mm^3 [/latex]

(total volume of cube)

[latex]\gamma_f =\left(1.26 ight) \left(9.81 {kN}/{m^3} ight) =12.36{kN}/{m^3}[/latex]

[latex]F_e = \gamma_f V_d-3.77N[/latex]

[latex]= \left(\frac{12.36 imes 10^3 N}{m^3} ight) \left(512 imes 10^3 mm^3 ight) \left(\frac{1m^3}{\left(10^3 mm ight)^3 } ight) -3.77N[/latex]

[latex]F_e = 6.33N - 3.77N = 2.56N[/latex]

Result A downward force of 2.56 N is required to hold the cube submerged in glycerin.

Question 5.3: A cube 80 mm on a side is made of a rigid foam material and ...

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