Neutron stars consist only of neutrons and have unbelievably high densities. A typical mass and radius for a neutron star might be 5.6 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)?
To find the density of the neutron star, we can use the formula:
Density = Mass / Volume
a) The given mass of the neutron star is 5.6 x 10^28 kg.
The volume can be calculated using the formula for the volume of a sphere:
Volume = (4/3) * π * Radius^3
Substituting the given radius of the neutron star (1.5 x 10^3 m) into the formula:
Volume = (4/3) * π * (1.5 x 10^3)^3 = 1.41 x 10^10 m^3
Now, we can calculate the density:
Density = Mass / Volume = 5.6 x 10^28 kg / 1.41 x 10^10 m^3 ≈ 3.96 x 10^18 kg/m^3
Therefore, the density of the neutron star is approximately 3.96 x 10^18 kg/m^3.
b) To calculate the weight of a dime made from this material, we need to know the weight of the material per unit volume. However, the given information does not specify this value. Without the weight per unit volume, we cannot make an accurate calculation for the weight of the dime.
To find the density of a neutron star, we need to divide its mass by its volume. Given that the mass of the neutron star is 5.6 x 10^28 kg and the radius is 1.5 x 10^3 m, we can calculate the volume using the formula for the volume of a sphere:
V = (4/3) * π * r^3
V = (4/3) * π * (1.5 x 10^3)^3
V ≈ 1.4137 x 10^10 m^3
(a) Now we can calculate the density by dividing the mass by the volume:
Density = Mass / Volume
Density ≈ 5.6 x 10^28 kg / 1.4137 x 10^10 m^3
Density ≈ 3.962 x 10^17 kg/m^3
So, the density of the neutron star is approximately 3.962 x 10^17 kg/m^3.
(b) To calculate the weight of the dime made from this material, we need to convert the volume of the dime to cubic meters:
Volume of the dime = 2.0 x 10^-7 m^3
Now we can use the density of the neutron star to find the weight of the dime:
Weight = Density * Volume
Weight ≈ 3.962 x 10^17 kg/m^3 * 2.0 x 10^-7 m^3
Weight ≈ 7.924 x 10^10 kg
To convert the weight to pounds, we need to know the conversion factor between kilograms and pounds. The conversion factor is approximately 2.205 pounds per kilogram.
Weight in pounds = Weight in kg * Conversion factor
Weight in pounds ≈ 7.924 x 10^10 kg * 2.205 pounds/kg
Weight in pounds ≈ 1.747 x 10^11 pounds
Therefore, the dime made from this neutron star material would weigh approximately 1.747 x 10^11 pounds.