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, we need to consider the forces acting on each object.
1. Beaker of oil:
The beaker is at rest and resting on a scale. The scale measures the weight of the beaker, including the mass of the oil it contains.
The weight of the beaker (W_b) is given by the formula:
W_b = m_b * g
where m_b is the mass of the beaker and g is the acceleration due to gravity.
2. Iron block submerged in oil:
The block of iron is completely submerged in the oil, so it will experience an upward buoyant force and a downward force due to gravity. The spring scale measures the tension force in the string, which is equal to the weight of the iron block.
The buoyant force (F_b) acting on the iron block is given by Archimedes' principle:
F_b = ρ_1 * V * g
where ρ_1 is the density of the oil, V is the volume of the block submerged in the oil, and g is the acceleration due to gravity.
The weight of the iron block (W_2) is given by:
W_2 = m_2 * g
where m_2 is the mass of the iron block and g is the acceleration due to gravity.
Now, to find the equilibrium readings, we can set up the equation by considering the net forces acting on each object:
For the beaker of oil:
W_b = m_b * g
For the iron block submerged in oil:
F_b - W_2 = 0
ρ_1 * V * g - m_2 * g = 0
To solve for the equilibrium readings, we need to know the relationship between the mass of the oil (m_1) and the volume of the block submerged in the oil (V). This requires information about the shapes and sizes of the beaker, oil, and iron block.