Answer
Density of the wood $=6.7\times10^{2}$ $kg/m^{3}$
Work Step by Step
Since the block is floating and it is undergoes no net acceleration, the weight of block $(W)$ is equal to the buoyant force $(F_{b})$ by water.
This can be written as:
$W=F_{b}$ . . . . . . . . . . . . . . . . . . . . . . .(1)
Since two-thirds of the volume $V$ is under water, buoyant force is given by:
$F_{b}=\frac{2}{3}V \rho_{w}g$
Substituting the known values and setting $F_{b}$ as $\frac{2}{3}V \rho_{w}g$ in the previous equation and solving gives:
$mg = \frac{2}{3}V \rho_{w} g$
$m=\frac{2}{3}V\times1000$ $kg/m^{3}$ . . . . . . . . . . . . . . . . . . (2)
Here the equation is in terms of mass and volume but we need that in terms of density of the block.
So writing mass in terms of volume gives:
$mass=density\times volume$
$m=\rho_{block}\times V$
Substituting the value of $m$ in equation (2) and solving gives:
$\rho_{block}\times V=\frac{2}{3}V\times1000$ $kg/m^{3}$
$\rho_{block}= \frac{2}{3}\times1000$ $kg/m^{3}= 666.66$ $kg/m^{3}=6.6666\times10^{2}$ $kg/m^{3}$
$\rho_{block}\approx 6.7\times10^{2}$ $kg/m^{3}$
So the density of the wood is $6.7\times10^{2}$ $kg/m^{3}$.
