To determine an athlete's body fat, she is weighed first in air and then again while she's completely underwater. It is found that she weighs 690 N when weighed in air and 36.0 N when weighed underwater. What is her average density?
m g = 690
mg - 1000 kg/m^3 * V g = 36
1000 V g = 690 - 36 = 654 N
g = 9.8
so
V = 654/(1000 g)
rho = m/V = 690/g / .654/g = 1055 kg/m^3
just a little heavier than water (must have expelled all air from her lungs to sink like that)
To find the athlete's average density, we can use the concept of buoyancy.
The buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
We know that the athlete's weight in air is 690 N and her weight underwater is 36.0 N.
The weight in air is equal to the sum of the athlete's actual weight and the buoyant force:
Weight in air = Actual weight + Buoyant force
690 N = Actual weight + Buoyant force
Similarly, the weight underwater is equal to the buoyant force:
Weight underwater = Buoyant force
36.0 N = Buoyant force
Now, we can subtract the weight underwater from the weight in air to find the buoyant force acting on the athlete:
Buoyant force = Weight in air - Weight underwater
Buoyant force = 690 N - 36.0 N
Buoyant force = 654 N
To find the average density, we use the formula:
Density = Mass / Volume
But since we only have information about the athlete's weight, we need to use a slight variation of the density formula:
Density = Weight / (Volume × gravitational acceleration)
The gravitational acceleration can be considered as 9.8 m/s².
So we have:
Density = Weight / (Volume × 9.8 m/s²)
To determine the volume of the athlete, we use the fact that the buoyant force on the athlete is the weight of the water displaced by the athlete.
Buoyant force = Density of the fluid × Volume × gravitational acceleration
Buoyant force = Weight of the water displaced
Plugging in the calculated buoyant force and the gravitational acceleration:
654 N = Density × Volume × 9.8 m/s²
Finally, to find the average density, rearrange the equation:
Density = Buoyant force / (Volume × gravitational acceleration)
Density = 654 N / (Volume × 9.8 m/s²)
Unfortunately, without the information about the volume, we cannot determine the athlete's average density.
To determine the athlete's average density, we need to use the concept of buoyancy. The difference in weight between the athlete in air and underwater is due to the buoyant force acting on her body.
The buoyant force is equal to the weight of the water displaced by the athlete's body. Therefore, we can calculate the volume of the water displaced using the difference in weight between the two measurements.
The weight of the athlete in air is 690 N, while the weight of the athlete underwater is 36.0 N. The difference between these two weights is equal to the buoyant force acting on the athlete, which is the weight of the water displaced.
Buoyant Force = Weight in Air - Weight Underwater
Buoyant Force = 690 N - 36.0 N
Buoyant Force = 654 N
The buoyant force is also equal to the weight of the water displaced, which can be found using the equation:
Buoyant Force = Density of Water × Volume of Water Displaced × g
Where g is the acceleration due to gravity (approximately 9.8 m/s²) and the density of water is approximately 1000 kg/m³.
Therefore, we can write:
654 N = (1000 kg/m³) × (Volume of Water Displaced) × (9.8 m/s²)
Simplifying the equation:
Volume of Water Displaced = (654 N) / [(1000 kg/m³) × (9.8 m/s²)]
Volume of Water Displaced ≈ 0.067 m³
Now we can calculate the average density of the athlete. The average density is given by:
Average Density = Mass of Athlete / Volume of Water Displaced
To find the mass of the athlete, we can use the equation:
Weight in Air = Mass of Athlete × g
Solving for the mass:
Mass of Athlete = Weight in Air / g
Mass of Athlete = 690 N / 9.8 m/s²
Mass of Athlete ≈ 70.41 kg
Using this value, we can now calculate the athlete's average density:
Average Density = Mass of Athlete / Volume of Water Displaced
Average Density = 70.41 kg / 0.067 m³
Average Density ≈ 1048.36 kg/m³
Therefore, the athlete's average density is approximately 1048.36 kg/m³.