A 0.73 kg beaker containing 2.17 kg of oil with a density of 916 kg/m3 rests on a scale.
A 1.92 kg block of iron is suspended from a spring scale and completely submerged in the oil. The density of iron is 7.86 103 kg/m3.
(a) Find the equilibrium reading of the top spring scale.
(b) Find the equilibrium reading of the bottom spring scale.
To solve this problem, we'll need to look at the buoyant force on the block by the oil.
Before we do that, though, let's start with our known forces on the top spring. We'll start by pointing out that our first force will be the weight of the block (let g = 10 for simplicity). Our positive axis will be in the downward direction for this scale:
F(w) = mg = 1.92*10 = 19.2 N
Now, we need to measure the bouyant force provided by the oil on the block. Buoyant force is given by the equation:
F(b) = dVg
Where d is the density of the fluid, V is the volume of displaced fluid, and g is gravitational acceleration. The problem has given to us the mass and density of iron. We can now find the volume of iron, and, hence, the volume of displaced fluid:
V = m/d = 1.92/(7.86*10^3) = 2.44*10^(-4) m^3
So, using the density provided in the problem, we can calculate our buoyant force on the block of iron:
F(b) = 916*(2.44*10^-4)*10 = 2.24 N
So, our total force on the upper spring is as follows:
F(total) = F(w) - F(b)
Remember, we subtract buoyant force because it is in the opposite direction of weight.
F(total) = 19.2 - 2.24 = 16.96 N
Now, we must find the total force on the bottom scale.
Again, let's consider the weight of our beaker and fluid first:
F(w) = 0.73(10) + 2.17(10) = 29.0 N
Now, we can use Newton's 3rd law to determine the effect of the buoyant force downward onto the bottom scale.
If we take the oil and beaker as a unit, that unit is applying an upward force of 2.24 N on the iron block. Therefore, the unit is receiving a 2.24 N downward force from the block. Therefore, our scale reading becomes:
F(total) = F(w) + F(b)
Buoyant force is the reaction force, so it will be in the same direction as the weight of the beaker and oil. That's why we add it this time.
F(total) = 29.0 + 2.24 = 31.24 N
There you have it! I hope that helped you out!