Notes on Derivation of Normal Force and Consistency of Expressions

Newton's Second Law

• Fundamental Principle: Newton's second law states that the force acting on an object equals the mass of that object multiplied by its acceleration.

$F = m imes a$

Derivation of Normal Force

1. Understanding the Scenario:

   • We have a cube with volume $V<em>0$ and density $p</em>1$ resting at the bottom of a tank filled with liquid of density $p_0$.

2. Weight of the Cube:

   • The weight of the cube is given by:
     $W<em>{cube} = m</em>{cube} imes g = V<em>0 imes p</em>1 imes g$
     where $g$ is the acceleration due to gravity.

3. Buoyant Force:

   • The buoyant force acting on the cube (upward) is given by Archimedes’ principle:
     $F<em>{buoyancy} = V</em>0 imes p_0 imes g$

4. Applying Newton's Second Law:

   • The cube is at rest, thus the net force equals zero. So, we can set up the equilibrium condition as follows:
     $N - W<em>{cube} + F</em>{buoyancy} = 0$
     where $N$ is the normal force exerted by the tank on the cube.

5. Solving for Normal Force:

   • Rearranging the equilibrium equation gives:
     $N = W<em>{cube} - F</em>{buoyancy}$
   • Substituting the expressions for weight and buoyancy:
     $N = V<em>0 imes p</em>1 imes g - V<em>0 imes p</em>0 imes g$
   • Factor out common terms:
     $N = V<em>0 imes g (p</em>1 - p_0)$

Consistency Justification

• The derived expression for the normal force $N$ indicates that it depends on the difference in densities between the cube and the liquid.
• If $p<em>1 > p</em>0$, then $N$ will be positive, meaning the tank exerts a force on the cube. Conversely, if $p<em>1 < p</em>0$, then the buoyant force would be greater, leading to a normal force of less than the weight of the cube, potentially indicating floating.
• This aligns with the claim made in part (a) that the behavior of objects in fluids depends critically on the relative densities involved.
• The derived equation is consistent in that it accounts for the influence of both the weight of the cube and the buoyancy it experiences in the liquid, confirming the expectations of fluid mechanics principles.