Determining Bowling Ball Volume, Density, and Buoyancy

Mass Measurement and Conversion Procedure

• Mass Recording Protocol:

  • Measure and record the mass of the bowling ball on the designated data page using units of pounds ($\text{lbs}$).

• Mass Conversion Procedure:

  • Convert the recorded mass from pounds ($\text{lbs}$) to grams ($\text{g}$) using the standard unit conversion factor ($1\,\text{lb} = 453.592\,\text{g}$).

Geometric Volume Determination of a Sphere

• Sphere Volume Formula:
  • The geometric volume ($V$) of a bowling ball is calculated using the formula for the volume of a sphere:

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

• Parameter Definitions and Standards:
  • The constant $\pi$ is set to $3.14$ for this calculation.
  • The variable $r$ represents the radius of the bowling ball.
  • The radius ($r$) is defined as one half of the diameter ($d$):

$r = \frac{1}{2} d$

• Measurement Units:
  • The final volume of the bowling ball must be expressed in cubic centimeters ($\text{cm}^3$).

Density Calculation and Buoyancy Determination

• Density Formula:
  • Calculate the density ($\text{density}$) of the bowling ball by dividing its converted mass in grams ($\text{g}$) by its volume in cubic centimeters ($\text{cm}^3$):

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Pure Water Reference Density:

  • The density of pure water is exactly $1.00\,\text{g/mL}$ (or $1.00\,\text{g/cm}^3$).

• Buoyancy Rules:

  • If the calculated density of the bowling ball is greater than $1.00\,\text{g/mL}$, the bowling ball will sink in pure water.
  • If the calculated density of the bowling ball is less than $1.00\,\text{g/mL}$, the bowling ball will float in pure water.

• Observation Recording:

  • Document the physical result of whether the bowling ball floated or sank in pure water directly on the data page.

Step-by-Step Solution for a Hypothetical 6-Pound Bowling Ball

• Problem Overview:

  • Determine the mass conversion, volume, density, and buoyancy behavior for a hypothetical bowling ball weighing $6\,\text{lbs}$.

• Step 1: Mass Conversion to Grams:

  • Convert mass from pounds to grams using the conversion factor $1\,\text{lb} = 453.592\,\text{g}$:

$\text{Mass} = 6\,\text{lbs} \times \frac{453.592\,\text{g}}{1\,\text{lb}} = 2721.552\,\text{g}$

• Step 2: Determine Sphere Radius and Volume:
  • Standard bowling ball diameter ($d$) is approximately $8.50\,\text{inches}$ ($21.59\,\text{cm}$).
  • Calculate radius ($r$):

$r = \frac{1}{2} \times 21.59\,\text{cm} = 10.795\,\text{cm}$

• Calculate volume ($V$) using $\pi = 3.14$:

$r^3 = (10.795\,\text{cm})^3 = 1257.85\,\text{cm}^3$

$V = \frac{4}{3} \times 3.14 \times 1257.85\,\text{cm}^3 = 5266.39\,\text{cm}^3$

• Step 3: Density Calculation:
  • Apply the density formula with mass $2721.552\,\text{g}$ and volume $5266.39\,\text{cm}^3$:

$\text{Density} = \frac{2721.552\,\text{g}}{5266.39\,\text{cm}^3} \approx 0.517\,\text{g/cm}^3$

• Step 4: Buoyancy Evaluation:
  • Compare the calculated density ($0.517\,\text{g/cm}^3$) to the density of pure water ($1.00\,\text{g/mL}$):

$0.517\,\text{g/cm}^3 < 1.00\,\text{g/mL}$

• Conclusion: Because its density is less than $1.00\,\text{g/mL}$, the $6\,\text{lb}$ bowling ball will float in pure water.
• Record Observation: Floated.