A beaker of mass mb containing oil of mass m1 and density ρ1 rests on a scale. A block of iron of mass m2 and density ρ2 is suspended from a spring scale and completely submerged in the oil. Determine the equilibrium readings of both scales. (Use rho_1 for ρ1, rho_2 for ρ2, m_1 for m1, m_2 for m2, m_b for mb, and g when necessary.)
To determine the equilibrium readings of both scales in this scenario, we need to analyze the forces acting on the objects. Let's start by calculating the buoyant force on the block of iron.
The buoyant force can be found using the equation: F_b = ρ_fluid * V * g, where ρ_fluid is the density of the fluid, V is the volume of the submerged object, and g is the acceleration due to gravity.
The volume of the submerged object can be calculated using its mass and density: V = m_block / ρ_block.
The mass of the block can be written as: m_block = m_iron = ρ_iron * V.
Substituting this back into the volume equation, we get: V = m_iron / ρ_iron.
Now, let's calculate the buoyant force: F_b = ρ_fluid * V * g.
Substituting V, we get: F_b = ρ_fluid * (m_iron / ρ_iron) * g.
Next, we need to consider the forces acting on the beaker of oil on the scale. The scale measures the weight of the beaker (including the oil), which is given by: F_scale = (m_beaker + m_oil) * g.
The mass of the oil is given as m_oil = ρ_oil * V_oil, where V_oil is the volume of the oil.
To determine the equilibrium readings of both scales, the forces acting on the block of iron and the beaker of oil must be equal.
Therefore, we have: F_scale = F_b.
(m_beaker + m_oil) * g = ρ_fluid * (m_iron / ρ_iron) * g.
Now, let's substitute the equations for m_oil and m_iron into the equation above:
(m_beaker + ρ_oil * V_oil) * g = ρ_fluid * (ρ_iron * V) * g.
The density of the oil, ρ_oil, cancels out in the equation, as ρ_oil = ρ_fluid (given).
Simplifying the equation further, we get:
(m_beaker + ρ_fluid * V_oil) = ρ_fluid * ρ_iron * V.
Finally, rearranging the equation to solve for the scale readings:
m_beaker = (ρ_fluid * ρ_iron * V - ρ_fluid * V_oil ).
m_block = ρ_iron * V.
So, the equilibrium readings of the spring scale for the beaker (m_beaker) and the block of iron (m_block) are:
m_beaker = ρ_fluid * ρ_iron * V - ρ_fluid * V_oil.
m_block = ρ_iron * V.
By substituting the known values for ρ_fluid, ρ_iron, V, and V_oil, you can calculate the equilibrium readings for both scales.