Question

The mass density of an object is its mass divided by its volume. Write and simplify an expression for the density of a sphere having a mass \(m\) and a volume equal to \(4 \pi r^{3} / 3\)

Short answer

The simplified expression for the density of a sphere with mass \(m\) and volume \(4 \pi r^{3} / 3\) is \(\rho = \frac{3m}{4 \pi r^{3}}\).

Step-by-step solution

Write the formula for density

The density, denoted as \( \rho \), of an object is calculated as the mass \( m \) divided by its volume \( V \). So, the formula for density is \( \rho = \frac{m}{V} \).

Substitute the volume of a sphere

In the given problem, the volume \( V \) of the sphere is \( \frac{4}{3} \pi r^{3} \). Substitute this into the density formula so it becomes \( \rho = \frac{m}{\frac{4}{3} \pi r^{3}} \).

Simplify the density expression

To simplify the expression, multiply both the numerator and the denominator by the reciprocal of the fraction in the denominator. The density becomes \( \rho = \frac{m}{1} \times \frac{3}{4 \pi r^{3}} = \frac{3m}{4 \pi r^{3}} \).

Key concepts

Density Formula

Understanding the density of a material is essential in physics and engineering as it defines how compact the mass of the substance is within its volume. The density, denoted by the Greek letter rho \(\rho\), is calculated by the formula:\
\[\rho = \frac{m}{V}\]
Here, \(m\) stands for mass, which is the amount of matter in the object, and \(V\) denotes volume, representing the space the object occupies. The density formula is universally applicable, whether it is a solid, like a sphere, a liquid, or a gas. Knowing how to work with this formula empowers students to calculate the density of different shapes and materials, providing vital information about their composition and behavior under various conditions.

For instance, when a material's volume is harder to measure directly—like that of an irregularly shaped object—calculating its mass and using its known density can assist in determining the volume. This makes the density formula an invaluable tool not only in theoretical problems but also in practical applications like materials science and engineering design.

Volume of a Sphere

The volume of a sphere, an essential concept in geometry, describes the amount of space that the sphere occupies. A sphere is a perfectly round 3-dimensional shape, and its volume is given by the formula:\
\[ V = \frac{4}{3} \pi r^{3} \]
Within this expression, \(r\) is the radius of the sphere—the distance from the center to any point on its surface—and \(\pi\) approximately equals to 3.14159, which is a fundamental constant in mathematics. To ease understanding, imagine that the volume is how much liquid it would take to fill the sphere completely. Calculating volume is fundamental in various fields, such as physics for understanding phenomena like buoyancy and engineering for material selection and structural evaluation.

Practical Implications

For students, visualizing a real-life example, such as a ball or a globe, can reinforce the concept. Knowing the volume of a sphere is critical when dealing with spherical objects in real-world scenarios, including calculating the capacity of spherical containers or the air volume inside a bubble.

Simplifying Expressions

Simplifying expressions is a mathematical process used to transform a complex or bulky expression into a simpler or more manageable form. It is similar to tidying up a room—grouping like items together and putting them in order. When simplifying an algebraic expression, the goal is to make it as straightforward as possible, oftentimes to facilitate solving equations or visually parsing the pertinent information at a glance.

For the density of a sphere, the simplification involves rearranging and reducing terms to their least complex form. This typically involves combining like terms, using arithmetic operations, factoring, and canceling out terms where possible. By simplifying, the expression becomes more accessible and easier to use, especially when substituting values to solve problems.

Understanding Fractions

One essential skill in simplifying expressions with fractions is to multiply or divide both numerator and denominator by the same number or expression (the reciprocal) to eliminate complex fractions or to make cancellation possible. Real-life scenarios, like comparing portions or adjusting recipes, rely heavily on the ability to simplify fractions effectively.