Answer:
The maximum mass that can hang without sinking is 33.44 kg.
1. Understand the problem:
We are asked to find the maximum mass that can hang from a 70-cm diameter Styrofoam sphere in water without sinking. We are given the Styrofoam density (32 kg/m³) and need to assume the hanging mass's volume is negligible compared to the sphere.
2. Identify relevant principles:
Buoyancy: An object submerged in a fluid experiences an upward force equal to the weight of the fluid displaced by the object.
Equilibrium: For the sphere to remain suspended, the upward buoyant force must balance the downward gravitational force (weight of the sphere + hanging mass).
3. Solve the problem:
a. Calculate the volume of the Styrofoam sphere:
Radius = diameter/2 = 70 cm / 2 = 35 cm
Volume of a sphere = (4/3)πr³ = (4/3)π(35 cm)³ ≈ 49655 cm³ = 0.049655 m³
b. Calculate the mass of the displaced water:
Mass of water displaced = Volume of water displaced * Density of water
The volume of water displaced is equal to the volume of the submerged sphere.
Density of water ≈ 1000 kg/m³
Mass of water displaced = 0.049655 m³ * 1000 kg/m³ ≈ 49.655 kg
c. Calculate the buoyant force:
Buoyant force = Weight of water displaced = 49.655 kg * 9.81 m/s² (acceleration due to gravity) ≈ 486.7 N
d. Calculate the weight of the sphere:
Weight of sphere = Density of sphere * Volume of sphere * Gravity
Weight of sphere = 32 kg/m³ * 0.049655 m³ * 9.81 m/s² ≈ 158.9 N
e. Calculate the maximum weight of the hanging mass:
The total weight (sphere + hanging mass) must equal the buoyant force for equilibrium.
Weight of hanging mass = Buoyant force - Weight of sphere
Weight of hanging mass = 486.7 N - 158.9 N ≈ 327.8 N
f. Convert force to mass:
Mass of hanging mass = Weight of hanging mass / Gravity
Mass of hanging mass = 327.8 N / 9.81 m/s² ≈ 33.44 kg
Therefore, the maximum mass that can hang without sinking is approximately 33.44 kg.
The probable question can be: Styrofoam has a density of 32 kg/m Part A For the steps and strategies involved in solving a similar problem, you may view a Video Tutor Solution What is the maximum mass that can hang without sinking from a 70-cm diameter Styrofoam sphere in water? Assume the volume of the mass is negligible compared to that of the sphere. Express your answer with the appropriate units. НА ? m = 12.21 kg Submit Previous Answers Request Answer X Incorrect; Try Again; 9 attempts remaining