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Chapter 1: Problem 61

A star is estimated to have a mass of \(2 \times 10^{36} \mathrm{~kg}\). Assuming it to be a sphere of average radius \(7.0 \times 10^{5} \mathrm{~km}\), calculate the average density of the star in units of grams per cubic centimeter.

Short Answer

Expert verified
The average density of the star is approximately 1.4 grams per cubic centimeter.

Step by step solution

01

Write down the given values

We know the mass of the star is \(2 \times 10^{36} \mathrm{~kg}\) and its average radius is \(7.0 \times 10^{5} \mathrm{~km}\).
02

Convert the mass to grams

Since 1 kg is equal to 1000 grams, we can convert the mass of the star to grams: $$(2 \times 10^{36} \mathrm{~kg}) \times (1000 \frac{\mathrm{g}}{\mathrm{kg}}) = 2 \times 10^{39} \mathrm{~g}$$
03

Convert the average radius to centimeters

Since 1 km is equal to \(10^{5}\) centimeters, we can convert the radius to centimeters: $$(7.0 \times 10^{5} \mathrm{~km}) \times (10^{5} \frac{\mathrm{cm}}{\mathrm{km}}) = 7.0 \times 10^{10} \mathrm{~cm}$$
04

Calculate the volume of the star

The star is assumed to be a sphere, so we will use the formula for the volume V of a sphere: $$V = \frac{4}{3} \pi r^3$$ Plugging in the radius, we get: $$V = \frac{4}{3} \pi (7.0 \times 10^{10} \mathrm{~cm})^3 = \frac{4}{3} \pi (3.43\times 10^{32} \mathrm{~cm^3})$$
05

Calculate the average density of the star

Using the density formula (Density = Mass / Volume), we can find the average density: $$\rho = \frac{Mass}{Volume} = \frac{2 \times 10^{39} \mathrm{~g}}{\frac{4}{3} \pi (3.43\times 10^{32} \mathrm{~cm^3})}$$ After calculating, we get the average density of the star: $$\rho \approx 1.4 \, \mathrm{g/cm^3}$$ The average density of the star is approximately 1.4 grams per cubic centimeter.

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