Question

A typical neutron star may have a mass equal to that of the Sun but a radius of only \(10.0 \mathrm{~km} .(a)\) What is the gravitational acceleration at the surface of such a star? \((b)\) How fast would an object be moving if it fell from rest through a distance of \(1.20 \mathrm{~m}\) on such a star?

Short answer

The gravitational acceleration at the surface of the star is approximately \(1.32 \times 10^{12} \mathrm{ms^{-2}}\) and the speed of the object after falling a distance of 1.2m on the star from rest is approximately \(4.8 \times 10^{6} \mathrm{ms^{-1}}\).

Step-by-step solution

Identify Given Parameters

Let's denote the mass of the star (M) as the mass of the Sun which is approximately \(1.98 \times 10^{30} \mathrm{~kg}\). The radius (r) of the star is given as \(10.0 \mathrm{~km}\), which is \(1.0 \times 10^{4} \mathrm{~m}\). The gravitational constant (G) is \(6.67 \times 10^{-11} \mathrm{Nm^2kg^{-2}}\).

Calculate Gravitational Acceleration (Part A)

The formula for the gravitational acceleration at the surface of an object is \(g = \frac{GM}{r^2}\). Substituting the known values, we get \(g = \frac{(6.67 \times 10^{-11})(1.98 \times 10^{30})}{(1.0 \times 10^{4})^2}\) which simplifies to approximately \(1.32 \times 10^{12} \mathrm{ms^{-2}}\).

Calculate Speed of Falling Object (Part B)

Utilize the equation of motion which is \(v^2 = u^2 + 2gs\), where v is the final velocity, u is the initial velocity, g is the acceleration due to gravity, and s is the distance. Given that the initial velocity (u) of the object is 0 (since it starts from rest), distance (s) = 1.2m and g = \(1.32 \times 10^{12} \mathrm{ms^{-2}}\), substituting these values will give \(v^2 = 0 + 2(1.32 \times 10^{12})(1.2)\). The final velocity (v) will be approximately \(4.8 \times 10^{6} \mathrm{ms^{-1}}\) when calculated.

Key concepts

Neutron Star Mass

Neutron stars are the remnants of massive stars that have ended their lives in supernova explosions. Despite having a mass that can be comparable to that of our Sun, neutron stars are incredibly dense, resulting in a radius of only about 10 kilometers. This is because their composition is primarily neutron-rich matter; the intense gravity of the star's core having crushed the atomic structure to the point where protons and electrons combine to form neutrons. Understanding the mass of neutron stars is crucial because it affects their gravitational pull, which in turn influences their surroundings in space, including binary star systems and the emission of gravitational waves.

Gravitational Constant

The gravitational constant, denoted by the symbol G, is a key player in the universal law of gravitation. It is the proportionality constant used in Newton's law of universal gravitation, and it appears in other important physics equations, such as those applied in gravimetric measurements and celestial mechanics. The value of the gravitational constant is approximately \(6.67 \times 10^{-11} \mathrm{Nm^2kg^{-2}}\), which indicates the strength of gravity in a specific measurement system. It's crucial to apply the correct value of G when performing calculations related to gravitational force or acceleration, as it ensures the accuracy of the results. When considering objects such as neutron stars, the gravitational constant enables us to calculate extreme gravitational accelerations, demonstrating the vast differences between gravitational forces in space compared to those we experience on Earth.

Equation of Motion

The equation of motion for an object under the influence of gravity provides a relationship between initial velocity, final velocity, acceleration, and distance covered. Specifically, the equation \(v^2 = u^2 + 2gs\) is a useful kinematic equation, with 'v' representing final velocity, 'u' the initial velocity, 'g' the acceleration due to gravity, and 's' the distance traveled by the object. When applying this equation to a falling object on a neutron star, we input the immense gravitational acceleration already calculated and the fact that the object started from rest, meaning its initial velocity is zero. Using these values, the equation allows us to determine the final velocity of the object after it has fallen through a certain distance. This showcases how the fundamental principles of motion provide a window into understanding extreme scenarios like objects falling on the surface of a neutron star.