Q41P - Chemistry: Molecular Nature Of Matter And Change

Question

A steel ball-bearing with a circumference of 32.5 mm weighs 4.20 g. What is the density of the steel in g/cm³ (V of a sphere = 4/3 πr³ ; circumference of a circle = 2πr)?

Step-by-Step Solution

Answer

Density of the given steel ball comes out to be 7.241g/cm³ after the calculation using the given data.

Step 1: Density and Volume Definition and Formula

Density is the amount or mass present per unit volume and volume is the space occupied. In the question we have been provided a steel ball and ball represents a sphere.

For a sphere volume is space occupied within the sphere and it is calculated using the formula,

$V=\frac{4}{3}\pi r^3$ 

And, $Density=\frac{Mass}{Volume}$

Step 2: Calculating Volume of the Steel Ball (Sphere)

For calculating volume first, we need to find out the radius.

As given, $circumference of circle=2\pi r=32.5mm$ 

$$\begin{gathered} r\left(radius\right)=\frac{Circumference}{2\pi }=\frac{32.5}{2\times 3.14}=5.175mm=0.5175cm \\ V=\frac{4}{3}\pi r^3=\frac{4}{3}\times 3.14\times {(0.5175)}^3=0.58c{m}^3 \end{gathered}$$

Step 3: Calculating Density of Steel Ball

As discussed, 

$Density=\frac{Mass}{Volume}=\frac{4.20g}{0.58c{m}^3}=7.241g/c{m}^3$

Hence, the density of the given steel ball is 7.241g/cm³.