A certain solid metal object weighs 60 lb when measured in the normal manner in air, but it has such an irregular shape that it is difficult to calculate its volume by geometry. Use the principle of buoyancy to calculate its volume and specific weight.

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Accepted Answer

It is given that the weight of the object is 60 lb. Now, using a setup similar to that in Fig. 5.5, we find its apparent weight while submerged in water to be 46.5 lb. Using these data and the procedure for analyzing buoyancy problems, we can find the volume of the object.

Now apply Step 2 of the procedure and draw the free-body diagram of the object while it is suspended in the water.

The free-body diagram of the object while it is suspended in the water should look like Fig. 5.6. In this figure what are the two forces[latex]F_e [/latex] and w?

We know that w = 60 lb, the weight of the object in air, and [latex]F_e =46.5 lb [/latex], the supporting force exerted by the balance shown in Fig. 5.5.

Now do Step 3 of the procedure.

Using [latex] \sum{F_v}=0 [/latex] we get

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

Our objective is to find the total volume V of the object. How can we get V into this equation?

We use this equation from Step 4a,

[latex]F_b = \gamma_f V[/latex]

where [latex] \gamma_f[/latex] is the specific weight of the water, [latex]62.4 {lb}/{ft^3}[/latex]

Substitute this into the preceding equation and solve for V.

You should now have

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

[latex] \gamma_f V + F_e -W =0[/latex]

[latex]\gamma_f V = w-F_e[/latex]

[latex]V=\frac{W-F_e}{\gamma_f} [/latex]

Now we can put in the known values and calculate V.

Result The result is [latex]V= 0.216ft^3[/latex] . This is how it is done:

[latex]V=\frac{W-F_e}{\gamma_f} =\left(60-46.5\right) lb\left(\frac{ft^3}{62.4lb} \right) =\frac{13.5ft^3}{62.4} =0.216ft^3[/latex]

Comment Now that the volume of the object is known, the specific weight of the material can be found.

[latex]\gamma = \frac{W}{V}=\frac{60lb}{0.216ft^3} =278{lb}/{ft^3}[/latex]

This is approximately the specific weight of a titanium alloy.

Question 5.2: A certain solid metal object weighs 60 lb when measured in t...

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