Neutron stars consist only of neutrons and have unbelievably high densities. A typical mass and radius for a neutron star might be 5.8 x 1028 kg and 1.5 x 103 m. (a) Find the density of such a star. (b) If a dime (V = 2.0 x 10-7 m3) were made from this material, how much would it weight (in pounds)?

(a) Divide 5.8*10^28 kg by the volume of the star. The volume is (4/3)*pi*r^3.

(b) Multiply the density you get in (a) my 2*10^-7 m^3. That will give you the mass in kg. For the weight (actually, the mass) in lb, assume 2.2 lb per kg.

To find the density of the neutron star, we can use the formula:

Density = Mass / Volume

Given:
Mass of the neutron star = 5.8 x 10^28 kg
Radius of the neutron star = 1.5 x 10^3 m

(a) To find the density, we need the volume of the neutron star. The volume of a sphere can be calculated using the formula:

Volume = (4/3) x π x (radius)^3

Substituting the given values:
Volume = (4/3) x 3.14 x (1.5 x 10^3)^3

Now that we have the volume, we can calculate the density by dividing the mass by the volume:

Density = 5.8 x 10^28 kg / [(4/3) x 3.14 x (1.5 x 10^3)^3]

Calculating this expression will give us the density of the neutron star.

(b) To find the weight of a dime made from this material, we need to calculate the mass of the dime and then convert it to pounds.

Given:
Volume of the dime = 2.0 x 10^-7 m^3

Using the density we calculated in part (a), we can find the mass of the dime:

Mass = Density x Volume

Now that we have the mass of the dime, we can convert it to pounds. 1 kilogram is approximately equal to 2.20462 pounds.

Weight (in pounds) = Mass of the dime (in kg) x 2.20462

Calculating these expressions will give us the weight of the dime made from the neutron star material in pounds.

To find the density of a neutron star, we need to divide its mass by its volume. Here's how you can calculate it:

(a) Density of the neutron star:
Density = Mass / Volume

Given:
Mass = 5.8 x 10^28 kg
Volume = (1.5 x 10^3 m)^3

First, let's calculate the volume of the neutron star:
Volume = (1.5 x 10^3 m)^3
= 3.375 x 10^9 m^3

Now, we can calculate the density:
Density = Mass / Volume
= 5.8 x 10^28 kg / 3.375 x 10^9 m^3

To express the density in a more convenient unit, we can convert kg/m^3 to g/cm^3:
1 kg = 1000 g
1 m^3 = 10^6 cm^3

Density = (5.8 x 10^28 kg / 3.375 x 10^9 m^3) * (1000 g / 1 kg) * (1 m^3 / 10^6 cm^3)
= (5.8 x 10^28 kg / 3.375 x 10^9 cm^3)

Now, let's simplify and calculate the value of the density:

Density = 5.8 x 10^28 / 3.375 x 10^9
= 1.719 x 10^19 g/cm^3

Therefore, the density of the neutron star is approximately 1.719 x 10^19 g/cm^3.

(b) To calculate how much a dime made from this material would weigh, we need to find the mass of the dime and convert it to pounds. Here's how:

Given:
Volume of dime (V) = 2.0 x 10^-7 m^3

Using the density we calculated before (1.719 x 10^19 g/cm^3), we can find the mass of the dime:
Mass = Density * Volume

First, convert the volume of the dime to cm^3:
1 m^3 = 10^6 cm^3

Volume = 2.0 x 10^-7 m^3 * (10^6 cm^3 / 1 m^3)
= 2.0 x 10^-7 * 10^6 cm^3
= 2.0 x 10^-1 cm^3

Now, we can calculate the mass:
Mass = 1.719 x 10^19 g/cm^3 * 2.0 x 10^-1 cm^3

Simplifying and calculating the mass:
Mass = 1.719 x 10^19 * 2.0 x 10^-1 g
= 3.438 x 10^18 g

To convert the mass to pounds, we can use the conversion factor:
1 kg = 2.20462 pounds

Mass (in pounds) = (3.438 x 10^18 g / 1000) * (1 kg / 2.20462 pounds)
= 3.438 x 10^15 pounds

Therefore, a dime made from the material of the neutron star would weigh approximately 3.438 x 10^15 pounds.

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