Certain neutron stars (extremely dense stars) are believed to be rotating at about 100 rev/s. If such a star has a radius of 10 km, what must be its minimum mass so that material on its surface remains in place during the rapid rotation?
Assuming gravity is the force holding it together,
F=GMassStar*mass2/radius^2 where mass2 is the chunk wanting to fall off.
But
F= mass2 v^2/r
set them equal, find massStar.
To determine the minimum mass that a neutron star must have in order for the material on its surface to remain in place during its rapid rotation, we can make use of the concept of centrifugal force.
Centrifugal force is the apparent outward force experienced by an object moving in a curved path. In the case of a rotating object, such as a neutron star, the centrifugal force pulls material away from the center of rotation.
The centrifugal force acting on the surface of the neutron star must be balanced by the gravitational force pulling the material inward. Otherwise, the material would be flung off the surface.
The equation that relates these forces is:
Centrifugal force = Gravitational force
The centrifugal force can be calculated using the equation:
Centrifugal force = mass × angular velocity² × radius
where:
mass is the mass of the material on the surface,
angular velocity is the rotational speed of the neutron star in radians per second,
and radius is the radius of the star.
The gravitational force can be calculated using the equation:
Gravitational force = (G × mass × M) / (radius²)
where:
G is the gravitational constant,
mass is the mass of the material on the surface,
M is the mass of the neutron star,
and radius is the radius of the star.
To find the minimum mass of the neutron star, we need to set the centrifugal force equal to the gravitational force and solve for mass:
mass × angular velocity² × radius = (G × mass × M) / (radius²)
Simplifying the equation, we get:
mass × (angular velocity² × radius³) = G × mass × M
mass cancels out on both sides of the equation:
angular velocity² × radius³ = G × M
Now, we can solve for the minimum mass by rearranging the equation:
M = (angular velocity² × radius³) / G
Given that the rotational speed (angular velocity) is 100 revolutions/s and the radius is 10 km, we can convert the angular velocity to radians per second (2π radians = 1 revolution) and the radius to meters (1 km = 1000 m). Additionally, we can use the known value of the gravitational constant G (6.67430 × 10⁻¹¹ N(m/kg)²) to calculate the minimum mass of the neutron star.
Plugging in the values:
M = ((100 rev/s)² × (10,000 m)³) / (6.67430 × 10⁻¹¹ N(m/kg)²)
By evaluating this expression, we can find the minimum mass required for the material on the neutron star's surface to remain in place during its rapid rotation.