Question

According to the Official Rules of Baseball, a baseball must have a circumference not more than 9.25 in or less than 9.00 in and a mass not more than 5.25 oz or less than 5.00 oz. What range of densities can a baseball be expected to have? Express this range as a single number with an accompanying uncertainty limit.

Short answer

The density range of a baseball can be expressed as \(\rho_{avg} \pm \rho_{uncertainty}\), where \(\rho_{avg}\) is the average density calculated from the minimum and maximum densities, and \(\rho_{uncertainty}\) is the uncertainty limit computed as half the difference between the maximum and minimum densities.

Step-by-step solution

Calculate the radius from the given circumference

To find the radius of the baseball, we can use the formula for the circumference of a circle: \(C = 2\pi r\), where \(C\) is the circumference and \(r\) is the radius. We need to find the radius for both the minimum and maximum allowed circumferences. For the minimum circumference (9.00 in): \[ r_{min} = \frac{C_{min}}{2\pi} = \frac{9.00}{2\pi} \] For the maximum circumference (9.25 in): \[ r_{max} = \frac{C_{max}}{2\pi} = \frac{9.25}{2\pi} \]

Calculate the volume of the baseball using the radius values

Now that we have the values for the radius of the baseball, we can calculate the volume for both minimum and maximum radius using the formula for the volume of a sphere: \(V = \frac{4}{3}\pi r^3\). For the minimum radius: \[ V_{min} = \frac{4}{3}\pi (r_{min})^3 = \frac{4}{3}\pi \left(\frac{9.00}{2\pi}\right)^3 \] For the maximum radius: \[ V_{max} = \frac{4}{3}\pi (r_{max})^3 = \frac{4}{3}\pi \left(\frac{9.25}{2\pi}\right)^3 \]

Calculate the density range

Now we can calculate the range of densities for the given mass range, using the formula for density: \(\rho = \frac{m}{V}\), where \(\rho\) is density, \(m\) is mass, and \(V\) is volume. Density for minimum mass (5.00 oz) and minimum volume (9.00 in): \[ \rho_{1} = \frac{m_{min}}{V_{min}} = \frac{5.00}{\frac{4}{3}\pi \left(\frac{9.00}{2\pi}\right)^3} \] Density for maximum mass (5.25 oz) and minimum volume (9.00 in): \[ \rho_{2} = \frac{m_{max}}{V_{min}} = \frac{5.25}{\frac{4}{3}\pi \left(\frac{9.00}{2\pi}\right)^3} \] Density for minimum mass (5.00 oz) and maximum volume (9.25 in): \[ \rho_{3} = \frac{m_{min}}{V_{max}} = \frac{5.00}{\frac{4}{3}\pi \left(\frac{9.25}{2\pi}\right)^3} \] Density for maximum mass (5.25 oz) and maximum volume (9.25 in): \[ \rho_{4} = \frac{m_{max}}{V_{max}} = \frac{5.25}{\frac{4}{3}\pi \left(\frac{9.25}{2\pi}\right)^3} \] We want to find the minimum and maximum density values from the calculated densities \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\), and \(\rho_{4}\).

Express the density range

To express the density range as a single number with an accompanying uncertainty limit, we calculate the average of the minimum and maximum densities and the difference between the maximum and minimum densities as the uncertainty limit. Average density: \[ \rho_{avg} = \frac{\rho_{min} + \rho_{max}}{2} \] Uncertainty limit: \[ \rho_{uncertainty} = \frac{\rho_{max} - \rho_{min}}{2} \] So, the density range of a baseball is \(\rho_{avg} \pm \rho_{uncertainty}\).

Key concepts

Sphere Volume

To find the volume of a sphere, you need to use the formula \(V = \frac{4}{3}\pi r^3\), where \(V\) represents volume and \(r\) is the radius of the sphere. This formula is crucial because it helps define the three-dimensional space that the sphere occupies. Volume calculations are essential for problems involving packing, material usage, and storage.

To compute the range of possible baseball volumes, you first need the minimum and maximum radii derived from the circumferences. Given these radii, apply each one to the volume formula to get the smallest and largest volumes possible for the baseball. This allows you to establish how much physical space the baseball could occupy under the specified conditions.

In practical terms:

• Calculate \(r_{min}\) and \(r_{max}\) from the circumference.
• Apply each to the volume formula \(V = \frac{4}{3}\pi r^3\).
• Resulting \(V_{min}\) and \(V_{max}\) provide the volume range of the baseball.

Understanding volume helps connect how physical dimensions translate into space occupation.

Circumference Formula

The circumference of a baseball is a critical measurement because it links to several physical properties of the ball. The formula for calculating a circle's circumference is \(C = 2\pi r\), where \(C\) is the circumference and \(r\) is the radius. This formula is pivotal to translating the baseball's given circumference into the radius, which in turn is needed for further calculations, like finding the volume.

In this exercise:

• Ensure to calculate both minimum circumference \(C_{min}\) and maximum circumference \(C_{max}\).
• Use these values to find the corresponding radii \(r_{min}\) and \(r_{max}\).

Once you know the radius, it acts as a gateway to other computations, such as volume. Calculating accurately shows how minute circumference changes affect the entire physical profile of a sphere. This interconnectedness highlights the importance of precision in measurements.

Mass and Volume Relationship

The relationship between mass and volume defines an object's density, using the formula \(\rho = \frac{m}{V}\), where \(\rho\) is density, \(m\) is mass, and \(V\) is volume. Density helps describe how compact or spread out the mass is within the object's physical space. For a baseball, which has a defined mass range and volume range, this property is super vital.

Consider these steps in computing density:

• Use the smallest mass \(m_{min}\) with the largest volume \(V_{max}\), giving you the lowest density \(\rho_{3}\).
• Do the same for mass \(m_{max}\) with the smallest volume \(V_{min}\) for highest density \(\rho_{2}\).

Assessing these extremes lets us address the full range of densities a baseball might have. For practical applications, understanding this relationship is crucial for material science, manufacturing, and quality control.

Uncertainty in Measurements

Uncertainty in any measurement stems from limitations in the apparatus or processes used to conduct these measurements. In the exercise, uncertainties affect both the calculated radii and resulting density values. Presenting data with an appropriate uncertainty level is critical to convey the precision and reliability of information.

To express baseball density with uncertainty:

• Calculate the average of minimum and maximum densities to find an overall central density \(\rho_{avg}\).
• Compute the uncertainty by finding the difference between max and min density divided by two \(\rho_{uncertainty}\).

This method of expressing values \(\rho_{avg} \pm \rho_{uncertainty}\) provides a comprehensive look at potential variations, ensuring honesty about the limitations and giving users a succinct, useful insight into the range and reliability of calculations.