6.21) Calculate the density of a person who floats in fresh water with 5.0% of her volume above the surface.
Note: Provide solution/formula in the answer. Thanks!
Dp/Dw = 0.95.
Dp = 0.95Dw = 0.95*1000kg/m^3 = 950 kg/m^3 = Density of the person.
To calculate the density of a person who floats in fresh water with 5.0% of her volume above the surface, you can use the concept of buoyancy.
The formula for density is:
Density = Mass / Volume
In this case, we need to consider the buoyancy force acting on the person and how it relates to her density. When an object floats in a fluid, the buoyancy force exerted on the object is equal to the weight of the fluid displaced by the object.
First, let's assume the person's total volume is V. Since 5.0% of her volume is above the surface, 95.0% of her volume is submerged in water.
The buoyancy force can be calculated as:
Buoyancy force = Weight of water displaced
Since density is defined as mass divided by volume, the mass of water displaced can be calculated as:
Mass of water displaced = Density of water x Volume of water displaced
= Density of water x (V x 0.95) (95.0% of her volume is submerged)
The buoyancy force is equal to the weight of the water displaced:
Buoyancy force = Mass of water displaced x Gravity
Since the person floats, the buoyancy force should equal her weight:
Buoyancy force = Weight of the person
Mass of water displaced x Gravity = Mass of the person x Gravity
Rearranging the equation, we get:
Mass of water displaced = Mass of the person x (Volume of water displaced / Total volume of person)
Now we can substitute these values into the density formula:
Density = Mass of the person / Total volume of person
However, since we have the equation for mass of water displaced, we can plug it in:
Density = (Mass of person x (Volume of water displaced / Total volume of person)) / Total volume of person
Simplifying the equation, we get:
Density = Mass of person x (Volume of water displaced / Total volume of person)^2
So, to calculate the density of the person, you need to know the mass of the person, the volume of water displaced, and the total volume of the person.
To calculate the density of a person who floats in fresh water with 5.0% of their volume above the surface, we can use the formula:
Density = Mass / Volume
First, we need to determine the volume of the person that is below the surface of the water. Since 5.0% of her volume is above the surface, we can subtract this percentage from her total volume. Let's call this volume V.
V = (100% - 5.0%) * V
V = 95.0% * V
Now, let's denote the density of fresh water as ρ_water, which is approximately 1000 kg/m^3.
The weight of the person is equal to the weight of the water displaced by the submerged part of the body, which can be calculated using the Archimedes' principle:
Weight = ρ_water * g * V
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since weight = mass * g, we can rearrange the equation to find the mass:
Mass = Weight / g
Now we can substitute the weight expression into the equation:
Mass = (ρ_water * g * V) / g
Mass = ρ_water * V
Finally, we can substitute the mass and volume into the density equation to find the person's density:
Density = Mass / V
Substituting the values:
Density = (ρ_water * V) / V
Density = ρ_water
Therefore, the density of a person who floats in fresh water with 5.0% of her volume above the surface is approximately 1000 kg/m^3, which is the same as the density of fresh water.