A person weights 685 N in air but only 497 N when standing in water up to the hips. Find (a) the volume of each of the person’s legs and (b) the mass of each leg, assuming they have a density that is 1.05 times the density of water.
To solve this problem, we can use the concept of buoyancy. Buoyancy is the force exerted by a fluid (in this case, water) on an object immersed in it.
Let's solve the problem step-by-step:
Step 1: Find the volume of the person's legs.
We can use Archimedes' principle to find the volume. According to the principle, the buoyant force experienced by an object immersed in a fluid is equal to the weight of the fluid displaced by the object.
The person experiences a loss of weight equal to 685 N - 497 N = 188 N when standing in water. This loss of weight is due to the buoyant force exerted by the water on the legs.
Buoyant force = weight of the fluid displaced
The weight of the fluid displaced is equal to the difference in weight: 188 N. Let's assume the density of water is ρ (density of water = 1000 kg/m³).
Therefore, the volume of the legs can be calculated using the formula:
Volume = (Weight of the fluid displaced) / (Density of the fluid)
Volume = (188 N) / (ρ)
Step 2: Calculate the density of the person's legs.
The problem states that the density of the person's legs is 1.05 times the density of water (ρ). Therefore, the density of the legs (ρ_legs) can be calculated as:
Density of legs = 1.05 * (Density of water)
Step 3: Calculate the mass of the legs.
The mass of the legs (m_legs) can be calculated using the formula:
Mass = Density * Volume
Now, let's put it all together to solve the problem:
(a) Volume of each of the person's legs:
Volume = (188 N) / (ρ)
(b) Mass of each of the person's legs:
Density of legs = 1.05 * (Density of water)
Mass = Density * Volume
Note: In order to get the final numerical values for (a) and (b), we would need to know the value of density of water (ρ) provided in the question.
To solve this problem, we can use Archimedes' principle, which states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
Given:
Weight of the person in air (W_air) = 685 N
Weight of the person in water (W_water) = 497 N
Density of water (ρ_water) = ρ
Density of the person's legs (ρ_legs) = 1.05 ρ (where ρ is the density of water)
(a) To find the volume of each leg:
We can use the difference in weight when the person is in water to determine the volume of the water displaced by the legs.
Buoyant force acting on the legs (B) = W_air - W_water
B = ρ_water * g * V_legs
Rearranging the equation, we can solve for the volume of the legs:
V_legs = (W_air - W_water) / (ρ_water * g)
Substituting the given values:
V_legs = (685 N - 497 N) / (ρ * g)
(b) To find the mass of each leg:
We can use the density and volume to find the mass of each leg.
Mass of each leg (m_legs) = ρ_legs * V_legs
Substituting the given values:
m_legs = (1.05 ρ * V_legs)
Now, all we need is the value of the acceleration due to gravity (g) to plug into the equations.
Assuming a standard value of g = 9.8 m/s^2, we can proceed to calculate the values.
Please note that the actual value of g might differ depending on the location.
Now, you can calculate the volume of each leg (V_legs) and the mass of each leg (m_legs) using the formulas and the given values.