The volume of a neutron is about 10−45 cubic meters. Suppose you packed the number of neutrons you found for problem 4 into a cube so that the neutrons touched edge to edge. How big would the cube be? (Hint: Problem 4 states that a neutron star has a mass of 3.4×1030 kg and each neutron has the mass 1.7×10−27 kg. This means the neutron star contains 3.4 × 1030/1.7 × 10−27 = 2 × 1057 neutrons. Use the given volume for a single neutron to find the volume V of 2×1057 neutrons in a neutron star. Now set this equal to V = X3 and solve for the size X of a cubical neutron star.)
To find the size of the cube, we can start by calculating the total volume of the neutrons in the neutron star.
Given:
Volume of a neutron = 10^(-45) cubic meters
Number of neutrons in the neutron star = 2 × 10^57
The total volume V of the neutrons in the neutron star can be calculated by multiplying the volume of a single neutron by the total number of neutrons:
V = (10^(-45)) * (2 × 10^57)
= 2 × (10^(-45)) * (10^57)
= 2 × 10^(12)
= 2 × 10^12 cubic meters
Now, we can set this volume equal to the volume of a cube, V = X^3, where X represents the size of the cube.
2 × 10^12 = X^3
To solve for X, we can take the cube root of both sides:
∛(2 × 10^12) = ∛(X^3)
X = ∛(2 × 10^12)
Calculating the cube root:
X ≈ 128.379 meters
Therefore, the cube that would contain all the neutrons in the neutron star would have a size of approximately 128.379 meters.
To solve this problem, we need to find the size of a cubical neutron star given that it contains 2×10^57 neutrons and each neutron has a volume of 10^-45 cubic meters.
1. Find the volume of 2×10^57 neutrons in the neutron star:
The volume of a single neutron is 10^-45 cubic meters.
Multiply this by the number of neutrons in the star: 10^-45 * 2×10^57 = 2×10^(57-45) = 2×10^12 cubic meters.
2. Set this volume equal to the volume of a cube: V = X^3, where X is the size of the cube in meters.
So, 2×10^12 = X^3.
3. Solve for the size X of the cube using this equation.
Take the cube root of both sides: cube root (2×10^12) = X.
On most calculators, you can simply input the number and select the cube root function.
In this case, the cube root of 2×10^12 is approximately 2828.4.
Therefore, the cube that would contain the number of neutrons in a neutron star, with the neutrons touching edge to edge, would have a size of approximately 2828.4 meters.