(a) 198 g
When the rock is submerged into the water, there are two forces acting on the rock:
- its weight, equal to
(m=mass, g=acceleration of gravity), downward
- the buoyant force, equal to
(
=mass of water displaced), upward
So the resultant force, which is the apparent weight of the rock (W'), is
![W'=W-B](https://tex.z-dn.net/?f=W%27%3DW-B)
which can be rewritten as
![m'g = mg-m_w g](https://tex.z-dn.net/?f=m%27g%20%3D%20mg-m_w%20g)
where m' is the apparent mass of the rock. Using:
m = 540 g
m' = 342 g
we find the mass of water displaced
![m_w = m-m'=540 g-342 g=198 g](https://tex.z-dn.net/?f=m_w%20%3D%20m-m%27%3D540%20g-342%20g%3D198%20g)
(b) ![1.98\cdot 10^{-4} m^3](https://tex.z-dn.net/?f=1.98%5Ccdot%2010%5E%7B-4%7D%20m%5E3)
If the rock is completely submerged, the volume of the rock corresponds to the volume of water displaced.
The volume of water displaced is given by
![V_w = \frac{m_w}{\rho_w}](https://tex.z-dn.net/?f=V_w%20%3D%20%5Cfrac%7Bm_w%7D%7B%5Crho_w%7D)
where
is the mass of the water displaced
is the density of the water
Substituting,
![V_w = \frac{0.198}{1000}=1.98\cdot 10^{-4} m^3](https://tex.z-dn.net/?f=V_w%20%3D%20%5Cfrac%7B0.198%7D%7B1000%7D%3D1.98%5Ccdot%2010%5E%7B-4%7D%20m%5E3)
And so this is also the volume of the rock.
(c) ![2727 kg/m^3](https://tex.z-dn.net/?f=2727%20kg%2Fm%5E3)
The average density of the rock is given by
![\rho = \frac{m}{V}](https://tex.z-dn.net/?f=%5Crho%20%3D%20%5Cfrac%7Bm%7D%7BV%7D)
where
m = 540 g = 0.540 kg is the mass of the rock
is its volume
Substituting into the equation, we find
![\rho = \frac{0.540 kg}{1.98\cdot 10^{-4}}=2727 kg/m^3](https://tex.z-dn.net/?f=%5Crho%20%3D%20%5Cfrac%7B0.540%20kg%7D%7B1.98%5Ccdot%2010%5E%7B-4%7D%7D%3D2727%20kg%2Fm%5E3)