A plastic sphere with a radius of 375mm is completely submerged

General
1

a plastic sphere with a radius of 375mm is completely submerged

2

To provide a comprehensive answer, let us analyze this scenario step by step while incorporating relevant physics and mathematics concepts. This problem seems to involve the topic of fluid mechanics, specifically buoyancy and other principles of Archimedes.

A Plastic Sphere Fully Submerged in a Fluid

If a plastic sphere with a radius of 375 mm is fully submerged in a fluid, there are multiple scientific dimensions we could explore:

  1. Volume of the Sphere
    Since the sphere is submerged, its volume will directly interact with the fluid. To calculate the volume of a sphere, we use the formula:

    V = \frac{4}{3} \pi r^3

    Where:

    • r is the radius of the sphere.

    Calculation:

    Given:

    • Radius, r = 375 mm = 0.375 m (converted to meters since SI units are preferred).
      Substituting in the formula:
    V = \frac{4}{3} \pi (0.375)^3

    First, calculate 0.375^3:

    0.375^3 = 0.052734375

    Substitute:

    V = \frac{4}{3} \pi (0.052734375)
    V \approx \frac{4}{3} \cdot 3.14159 \cdot 0.052734375
    V \approx 0.2209 \, \text{m}^3

    Thus, the volume of the sphere is approximately 0.221 m³.

  1. Buoyant Force Acting on the Sphere
    The buoyant force exerted by the fluid on the submerged sphere can be calculated using Archimedes’ Principle:

    F_b = \rho_f V g

    Where:

    • \rho_f is the density of the fluid (in kg/m³),
    • V is the volume of the submerged sphere (calculated above as 0.221 \, \text{m}^3),
    • g is the gravitational acceleration (9.81 \, \text{m/s}^2).

    Calculation:

    Let us assume the sphere is submerged in water, where the density of water is:

    • \rho_f = 1000 \, \text{kg/m}^3.

    Substitute:

    F_b = 1000 \cdot 0.221 \cdot 9.81
    F_b \approx 2167.01 \, \text{N}

    So, the buoyant force acting on the sphere is approximately 2167 N.

  1. Weight of the Plastic Sphere
    To understand whether the sphere will float or sink, we need to compare the buoyant force (F_b) with the weight of the sphere (W). The weight of the sphere can be calculated using Newton’s second law:

    W = \rho_s V g

    Where:

    • \rho_s is the density of the plastic sphere (in kg/m³),
    • V is the volume of the sphere (calculated as 0.221 \, \text{m}^3),
    • g is the gravitational acceleration (9.81 \, \text{m/s}^2).

    Determining the Sphere’s Density:

    Typically, the density of plastic can vary widely depending on the type of plastic. For example:

    • Low-density plastics like expanded polystyrene can have densities around 10 \, \text{kg/m}^3,
    • Medium-density plastics like polypropylene can be around 900 \, \text{kg/m}^3,
    • High-density plastics like acrylics can exceed 1100 \, \text{kg/m}^3.

    Let us calculate the weight for a medium-density plastic (900 \, \text{kg/m}^3):

    Substitute:

    W = 900 \cdot 0.221 \cdot 9.81
    W \approx 1952.56 \, \text{N}

    Thus, the weight of the sphere is approximately 1953 N.

  1. Will the Sphere Sink or Float?
    Now, compare the buoyant force (F_b) and the weight of the sphere (W):

    • Buoyant Force: 2167 N,
    • Weight of Sphere: 1953 N.

    Since the buoyant force is greater than the weight of the plastic sphere, the sphere will float when fully submerged.

  1. Additional Analysis – How Much of the Sphere Will Be Above the Surface?
    If the sphere floats, part of it will be above the water surface. To calculate the fraction of the sphere that will remain submerged (also called volume fraction submerged):

    \text{Fraction Submerged} = \frac{\text{Weight of Sphere}}{\text{Buoyant Force}}

    Substituting:

    \text{Fraction Submerged} = \frac{1953}{2167}
    \text{Fraction Submerged} \approx 0.901

    Approximately 90.1% of the sphere’s volume will remain submerged in water, and the remaining 9.9% will stay above the water surface.

Conclusion:

  • Volume of the sphere: 0.221 \, \text{m}^3.
  • Buoyant force: 2167 \, \text{N}.
  • Weight of the plastic sphere: 1953 \, \text{N}.
  • Behavior: The sphere will float on water.
  • Fraction submerged: Approximately 90.1%.

If more details are available (e.g., the exact type/density of plastic or the fluid’s density), these calculations can be refined further. Let me know if you need elaboration on any specific part! :blush: