How much of the wooden timber was submerged in water?

AI Thread Summary
The discussion revolves around calculating how much of a wooden timber, with a mass of 20 g and a density of 0.27 g/cc, is submerged in water with a density of 0.970 g/cc. The user initially applies the principles of buoyancy and weight, leading to a volume calculation of 20.6185 m³, which is incorrect due to a unit conversion error. The correct volume, as per the book, is 20.62 cc, highlighting a misunderstanding of mass conversion from grams to kilograms. The user realizes the mistake was in misapplying the conversion factor, leading to the confusion in the final volume calculation. This emphasizes the importance of accurate unit conversions in physics problems.
Istiak
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Homework Statement
Mass of a timber is $20 \ g$. And, density of that timber is $0.27 \ g/cc$. That timber was bind to a metallic materials and, it was released to $0.970 \ g/cc$ water. How much the wood was submerged in water?
Relevant Equations
$$F=Ah\rho g$$
>Mass of a timber is $20 \ g$. And, density of that timber is $0.27 \ g/cc$. That timber was bind to a metallic materials and, it was released to $0.970 \ g/cc$ water. How much the wood was submerged in water?

I was trying to solve the problem following way.

$$F=Ah\rho g$$
$$=V\rho g$$
$$=V \ m^3 \cdot 970 \ kgm^{-3} \cdot 9.8 ms^{-2}$$
$$=9506V \ N$$
$$W=mg$$
$$=20_{\times 10^3} \ kg \times 9.8 \ ms^{-2}$$
$$=196000 \ N$$
$$W=F$$
$$9506V \ N=196000 \ N$$
$$V=20.6185 \ m^3$$

I think that I didn't do any mistake while solving that problem. Even, I didn't do any mistake of Units. But, I was wondering why my book wrote that $V=20.62 \ cc$. Both answer matched. But, I was thinking of Units. I am just showing a line what they did than, you will understand what they actually did.

$$19620 \ dyne = 951.57 V \ dyne$$
$$V=20.62 \ cm^3=20.62 \ cc$$
 
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Istiakshovon said:
>Mass of a timber is $20 \ g$.

Istiakshovon said:
$$=20_{\times 10^3} \ kg$$
Do you see the factor of ##10^6## error there?
 
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jbriggs444 said:
Do you see the factor of 10^6 error there?
Ohh! $1 \ kg = 1000 \ g$ but, I took opposite. That's the mistake than,
 
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