The mass of a certain neutron star is 1 × 1031 kg (5 solar masses) and its radius is 5500 m.
What is the acceleration of gravity at the surface of this condensed, burned-out star? The value of the universal gravitational con- stant is 6.67 × 10−11 N · m2/kg2.
Answer in units of m/s2
To calculate the acceleration of gravity at the surface of the neutron star, we can use the formula for gravitational acceleration:
\[ g = \frac{{G \cdot M}}{{r^2}} \]
where:
- g is the acceleration of gravity
- G is the universal gravitational constant (6.67 × 10^-11 N · m^2/kg^2)
- M is the mass of the neutron star
- r is the radius of the neutron star
Given:
- M = 1 × 10^31 kg
- r = 5500 m
Substituting the given values into the formula, we get:
\[ g = \frac{{(6.67 × 10^{-11} \, N \cdot m^2/kg^2) \cdot (1 × 10^{31} \, kg)}}{{(5500 \, m)^2}} \]
Now we can calculate the acceleration of gravity:
\[ g = \frac{{6.67 × 10^{-11} \cdot 1 × 10^{31}}}{{5500^2}} \, \frac{{N \cdot m^2/kg^2}}{{m^2}} \]
First, let's calculate the value in scientific notation:
\[ g = \frac{{6.67 \cdot 1}}{{5500^2}} \cdot 10^{(−11 + 31)}} \, \frac{{N}}{{kg}} \]
\[ g = \frac{{6.67 \cdot 1}}{{5500^2}} \cdot 10^{20} \, \frac{{N}}{{kg}} \]
Now let's calculate the value within the parentheses:
\[ g = \frac{{6.67 \cdot 1}}{{5500^2}} \times 10^{20} \, \frac{{N}}{{kg}} \]
\[ g = \frac{{6.67}}{{5500^2}} \times 10^{20} \, \frac{{N}}{{kg}} \]
Evaluate the numerator:
\[ g = \frac{{6.67}}{{(5.5 \times 10^3)^2}} \times 10^{20} \, \frac{{N}}{{kg}} \]
Evaluate the denominator:
\[ g = \frac{{6.67}}{{3.025 \times 10^7}} \times 10^{20} \, \frac{{N}}{{kg}} \]
Now divide to find the value of g:
\[ g = 2.200 \times 10^{−18} \, \frac{{N}}{{kg}} \]
Therefore, the acceleration of gravity at the surface of the neutron star is approximately 2.200 × 10^-18 m/s^2.