The maximum mass that can hang without sinking from the 50-cm diameter Styrofoam sphere in water is approximately 63.3 kg.**
## Finding the maximum hanging mass for a Styrofoam sphere
We can solve this problem by considering the buoyant force acting on the sphere and comparing it to the weight of the hanging mass.
**Step 1: Calculate the volume of the Styrofoam sphere.**
- Diameter = 50 cm = 0.5 m
- Radius = 0.5 m / 2 = 0.25 m
- Volume of a sphere = (4/3)πr³, so V_sphere = (4/3)π * 0.25³ ≈ 0.0654 m³
**Step 2: Calculate the mass of the sphere.**
- Density of Styrofoam = 32 kg/m³
- Mass of sphere = Density * Volume = 32 kg/m³ * 0.0654 m³ ≈ 2.10 kg
**Step 3: Calculate the buoyant force acting on the sphere.**
- Buoyant force = Weight of water displaced by the submerged portion of the sphere
- Since the entire sphere is submerged, the buoyant force equals the weight of the water displaced by the entire sphere's volume.
- Density of water = 1000 kg/m³
- Weight of water displaced = Density of water * Volume of water displaced = 1000 kg/m³ * 0.0654 m³ ≈ 65.4 kg
**Step 4: Determine the maximum hanging mass.**
- The sphere will remain afloat as long as the buoyant force is greater than or equal to the combined weight of the sphere and the hanging mass.
- Therefore, the maximum hanging mass is:
Maximum hanging mass = Buoyant force - Weight of sphere = 65.4 kg - 2.10 kg ≈ 63.3 kg
**Therefore, the maximum mass that can hang without sinking from the 50-cm diameter Styrofoam sphere in water is approximately 63.3 kg.**