A 77.8-kg person puts on a life jacket, jumps into the water, and floats. The jacket has a volume of 3.05 x 10-2 m³ and is completely submerged under the water. The volume of the person’s body that is underwater is 6.12 x 10-2 m³. What is the density of the life jacket? Number IM Units
The Correct Answer and Explanation is:
Answer:
The density of the life jacket is:
184.2 kg/m³
Explanation
To determine the density of the life jacket, we can use the principle of buoyancy, specifically Archimedes’ Principle, which states:
The buoyant force acting on a submerged object is equal to the weight of the fluid (in this case, water) it displaces.
Step 1: Total Buoyant Force
The person floats, meaning the upward buoyant force equals their weight.
The volume of water displaced = volume of life jacket + volume of submerged body Vdisplaced=3.05×10−2+6.12×10−2=9.17×10−2 m3V_{\text{displaced}} = 3.05 \times 10^{-2} + 6.12 \times 10^{-2} = 9.17 \times 10^{-2} \, \text{m}^3
Since the density of water is approximately 1000 kg/m³: Fbuoyant=ρwater⋅g⋅Vdisplaced=1000⋅9.8⋅9.17×10−2=898.66 NF_{\text{buoyant}} = \rho_{\text{water}} \cdot g \cdot V_{\text{displaced}} = 1000 \cdot 9.8 \cdot 9.17 \times 10^{-2} = 898.66 \, \text{N}
The person’s weight is: W=m⋅g=77.8⋅9.8=761.44 NW = m \cdot g = 77.8 \cdot 9.8 = 761.44 \, \text{N}
This confirms the person floats because the buoyant force (898.66 N) is greater than their weight (761.44 N).
Step 2: Determine Mass of Life Jacket
The life jacket provides the additional buoyant force beyond the person’s weight: Fjacket=Fbuoyant−W=898.66−761.44=137.22 NF_{\text{jacket}} = F_{\text{buoyant}} – W = 898.66 – 761.44 = 137.22 \, \text{N}
This force comes from the life jacket’s displacement, so we find its mass: mjacket=Fjacketg=137.229.8=14.01 kgm_{\text{jacket}} = \frac{F_{\text{jacket}}}{g} = \frac{137.22}{9.8} = 14.01 \, \text{kg}
Step 3: Calculate Density
ρjacket=mjacketVjacket=14.013.05×10−2=459.0 kg/m3\rho_{\text{jacket}} = \frac{m_{\text{jacket}}}{V_{\text{jacket}}} = \frac{14.01}{3.05 \times 10^{-2}} = \boxed{459.0} \, \text{kg/m}^3
Correction: Wait! We made a mistake assuming the jacket provides the net buoyant force. Actually, the jacket has its own mass and contributes to the total weight, so let’s use this refined method:
Let mjm_j = mass of life jacket
Total mass: mperson+mjm_{\text{person}} + m_j
Total weight = (77.8+mj)⋅g(77.8 + m_j) \cdot g
So, the total buoyant force: ρwater⋅g⋅Vtotal=(77.8+mj)⋅g\rho_{\text{water}} \cdot g \cdot V_{\text{total}} = (77.8 + m_j) \cdot g 1000⋅(9.17×10−2)=77.8+mj⇒91.7=77.8+mj⇒mj=13.9 kg1000 \cdot (9.17 \times 10^{-2}) = 77.8 + m_j \Rightarrow 91.7 = 77.8 + m_j \Rightarrow m_j = 13.9 \, \text{kg} ρjacket=13.93.05×10−2=456.1 kg/m3\rho_{\text{jacket}} = \frac{13.9}{3.05 \times 10^{-2}} = \boxed{456.1 \, \text{kg/m}^3}
🟩 Final Answer: 456.1 kg/m³
Let me know if you’d like this in a clean format for a report.
