Answer:
[tex]\displaystyle V_l=V\ \frac{\rho_o -\rho_w }{\rho_l-\rho_w}[/tex]
Explanation:
Density
It's a physical magnitude that relates the mass of an object with the volume it occupies. The formula is
[tex]\displaystyle \rho = \frac{m}{V}[/tex]
Equivalently
[tex]m=\rho V[/tex]
The density of water is 1 gr/ml, 1 Kg/lt, 1000 kg/m^3 or any equivalent unit.
Assume the wood sphere has a volume V and a density [tex]\rho_w[/tex], thus
[tex]m_w=\rho_w V[/tex]
A hole is to be drilled inside the wood so it means part of its volume will be filled with lead, which mass will be
[tex]m_l=\rho_l V_l[/tex]
The volume of wood is the total volume V minus the (unknown) volume of lead, thus
[tex]V_w=V-V_l[/tex]
The total mass of the modified sphere is
[tex]m_s=m_w+m_l=\rho_w V_w+\rho_l V_l[/tex]
Substituting Vw
[tex]m_s=m_w+m_l=\rho_w (V-V_l)+\rho_l V_l[/tex]
Operating
[tex]m_s=m_w+m_l=\rho_w V-\rho_w V_l+\rho_l V_l[/tex]
[tex]m_s=m_w+m_l=\rho_w V+ V_l(\rho_l -\rho_w)[/tex]
The total volume of the sphere doesn't change, which means
[tex]V_s=V[/tex]
The new density of the modified sphere is
[tex]\displaystyle \rho_s = \frac{m_s}{V}[/tex]
[tex]\displaystyle \rho_s = \frac{\rho_w V+ V_l(\rho_l -\rho_w)}{V}[/tex]
This density must be equal to the density of water [tex]\rho_o[/tex]
[tex]\displaystyle \rho_o = \frac{\rho_w V+ V_l(\rho_l -\rho_w)}{V}[/tex]
Operating and solving for vl
[tex]\displaystyle V_l=\frac{\rho_o V-\rho_w V}{\rho_l-\rho_w}[/tex]
Or equivalently
[tex]\displaystyle \boxed{V_l=V\ \frac{\rho_o -\rho_w }{\rho_l-\rho_w}}[/tex]
The equation would be enough for the volume of wood to keep the sphere neutrally buoyant would be [tex]Vt= Vp0-Pw/Pt-Pw[/tex]
The formula for the physical magnitude is p= m/v
Given that the density of water is 1 gr/ml, 1 Kg/lt, 1000 kg/m^3 or any equivalent unit.
We would assume the wood sphere has a volume V and a density Pw,
[tex]Mw= PwV[/tex]
Since a hole is to be drilled inside the wood so it means part of its volume will be filled with lead, the mass will be [tex]Mt= PtVt[/tex]
The volume of wood is the total volume V minus the (unknown) volume of lead, thus
[tex]Vw= V- Vt[/tex]
Hence, after further evaluation, the density would be [tex]Vt= V Po- Pw/Pt- Pw[/tex]
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