A cylindrical log of uniform density and radius 20.0
cm floats so that the vertical distance from the water
line to the top of the log is 8.00 cm. What is the density
of the log? (Hint: draw a good picture of the circular
end of the log.)
Dia. = 2 * 20 = 40 cm.
hb = height below the surface.
Dw = 1g/cm^3 = The density of water.
hb = Dl/Dw * h = 40-8 = 32 cm.
Dl/1 * 40 = 32, Dl = 0.8g/cm^3.
To determine the density of the cylindrical log, we can use the concept of buoyancy. Since the log is floating vertically in water, the buoyant force upward must balance the weight of the log downward.
Let's start by considering the log immersed in water. When the log is in equilibrium, the weight of the log (W_log) should be equal to the buoyant force (F_buoyant) acting on it.
The weight of the log can be calculated using the formula:
W_log = m * g
Where:
- W_log is the weight of the log
- m is the mass of the log
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Since the log has a uniform density, its mass will be proportional to its volume. The volume of a cylinder is given by the formula:
V = π * r^2 * h
Where:
- V is the volume of the cylinder
- r is the radius of the log
- h is the height of the log (given as 8.00 cm or 0.08 m)
Now, let's consider the buoyant force acting on the log. The buoyant force can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object.
The volume of water displaced by the log is equal to the volume of the submerged portion, which is a cylinder with the same radius (r) but a different height. The height of the submerged portion of the log is equal to the vertical distance from the water line to the top of the log, given as 8.00 cm or 0.08 m.
Using the same formula for the volume of a cylinder, we can calculate the volume of the submerged portion.
V_submerged = π * r^2 * h_submerged
Given that h_submerged = 0.08 m, we can substitute it into the equation.
Now that we have the volume of the submerged portion, we can calculate the buoyant force as:
F_buoyant = ρ_water * V_submerged * g
Where:
- F_buoyant is the buoyant force
- ρ_water is the density of water (approximately 1000 kg/m^3)
Since the log is in equilibrium, we can set the weight of the log equal to the buoyant force:
m * g = ρ_water * V_submerged * g
We can then cancel out the g terms:
m = ρ_water * V_submerged
Since we want to find the density of the log, we can substitute the formula for mass (m = ρ * V) into the equation:
ρ * V = ρ_water * V_submerged
Rearranging the equation, we can solve for the density of the log:
ρ = (ρ_water * V_submerged) / V
Substituting the formulas for the volume of the cylinder and the submerged portion, we get:
ρ = (ρ_water * π * r^2 * h_submerged) / (π * r^2 * h)
We can now substitute the given values:
- r = 20.0 cm or 0.2 m
- h_submerged = 0.08 m
- h = 0.08 m
- ρ_water = 1000 kg/m^3
After substituting these values into the equation, we can calculate the density of the log.