Answer
(a) The buoyant force is $8.82~N$
(b) The buoyant force is $9.62~N$
Work Step by Step
(a) If the ice is floating freely in the water, then the buoyant force is equal in magnitude to the weight of the ice.
We can find the buoyant force:
$F_b = mg = (0.90~kg)(9.80~m/s^2) = 8.82~N$
The buoyant force is $8.82~N$
(b) According to Archimedes' principle, the buoyant force is equal to the weight of the water that is displaced.
Let $m_i$ be the mass of the ice. Let $m_w$ be the mass of the water that is displaced by the ice. Let $V$ be the volume of the ice. The buoyant force is equal to the weight of the water with a volume $V$:
$F_b = m_w~g$
$F_b = \rho_w~V~g$
$F_b = \rho_w~(\frac{m_i}{\rho_i})~g$
$F_b = \frac{(1000~kg/m^3)(0.90~kg)(9.80~m/s^2)}{917~kg/m^3}$
$F_b = 9.62~N$
The buoyant force is $9.62~N$.
