When a block of volume 1.00 × 10–3 m3
is hung from a spring scale,
the scale reads 10.0 N. When the same block is then placed in an unknown liquid, it floats with 2/3 of its volume submerged. The density of water is 1.00 × 103 kg/m^3. Determine the density of the unknown liquid.
10
180
To determine the density of the unknown liquid, we can use Archimedes' principle. According to Archimedes' principle, the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
Let's break down the problem into steps:
Step 1: Find the buoyant force acting on the block.
The buoyant force (Fb) is given by the formula:
Fb = ρvlg
Where:
ρ = Density of the fluid (unknown liquid)
v = Volume of the fluid displaced by the block (2/3 of the volume of the block)
l = Acceleration due to gravity (assuming 9.8 m/s^2)
g = Volume of the block (given as 1.00 × 10–3 m^3)
Step 2: Find the weight of the block in the fluid.
The weight of the block (W) can be determined from the reading on the spring scale (10.0 N).
Step 3: Equate the buoyant force and the weight of the block.
Since the block is in equilibrium, the buoyant force should be equal to the weight of the block.
Fb = W
Step 4: Solve for the density of the unknown liquid.
Substitute the values into the formula and solve for ρ (density of the unknown liquid).
Now, let's go through each step in detail:
Step 1:
Fb = ρvlg
Fb = (ρ)(2/3 × 1.00 × 10–3 m^3)(9.8 m/s^2)
Step 2:
W = 10.0 N
Step 3:
Fb = W
Step 4:
Substitute the values from step 1, 2, and 3 into the equation and solve for ρ:
(ρ)(2/3 × 1.00 × 10–3 m^3)(9.8 m/s^2) = 10.0 N
Simplify the equation:
(ρ)(2/3)(1.00 × 10–3 m^3)(9.8 m/s^2) = 10.0 N
Rearrange the equation to solve for ρ:
ρ = (10.0 N)/((2/3)(1.00 × 10–3 m^3)(9.8 m/s^2))
Calculate the value of ρ using a calculator:
ρ ≈ 821.14 kg/m^3
Therefore, the density of the unknown liquid is approximately 821.14 kg/m^3.
To determine the density of the unknown liquid, we need to consider the buoyant force acting on the block when it is submerged in the liquid.
Buoyant force (F_b) is given by the equation:
F_b = ρ_l * g * V_disp
Where:
- F_b is the buoyant force,
- ρ_l is the density of the liquid,
- g is the acceleration due to gravity,
- V_disp is the volume of the liquid displaced by the submerged portion of the block.
From the problem statement, we know that the volume of the block is 1.00 × 10^(-3) m^3 and 2/3 of its volume is submerged. Hence, the volume of the liquid displaced is (2/3) * (1.00 × 10^(-3)) m^3.
The gravitational force (weight) acting on the block is given by:
F_g = m * g
Where:
- F_g is the weight of the block,
- m is the mass of the block,
- g is the acceleration due to gravity.
We can equate the weight of the block to the scale reading of 10.0 N to find the mass of the block.
Since weight is given by:
F_g = m * g
And the scale reading is 10.0 N, we have:
10.0 N = m * g
Using the value of g as 9.81 m/s^2, we can solve for the mass (m) of the block.
Once we have the mass of the block, we can calculate the density of the unknown liquid by rearranging the equation for buoyant force:
ρ_l = F_b / (g * V_disp)
Now, let's calculate step by step:
1. Calculate the mass of the block:
F_g = m * g
10.0 N = m * 9.81 m/s^2
Solve for m.
2. Calculate the volume of the liquid displaced:
V_disp = (2/3) * (1.00 × 10^(-3)) m^3
3. Calculate the buoyant force:
F_b = ρ_l * g * V_disp
4. Substitute the known values into the equation and solve for ρ_l:
ρ_l = F_b / (g * V_disp)
5. Calculate the density of the unknown liquid.