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Certain neutron stars (extremely dense stars) are believed to be rotating at about 2.0 rev/s. If such a star has a radius of 34 km, what must be its minimum mass so that material on its surface remains in place during the rapid rotation?
Number
Units
To calculate the minimum mass of the neutron star required for material on its surface to stay in place during its rapid rotation, we can use the concept of centripetal acceleration and the gravitational force acting on the star.
1. Centripetal Acceleration Formula: For an object moving in a circle, centripetal acceleration ( a_c ) is given by:
[
a_c = frac{v^2}{r}
]
where ( v ) is the tangential velocity and ( r ) is the radius.
2. Tangential Velocity: Since the star is rotating at 2.0 revolutions per second, we can find the tangential velocity using:
[
v = 2 pi r f
]
where ( f ) is the frequency of rotation (2.0 rev/s).
3. Gravitational Force: For an object to remain on the surface without flying off due to rotation, the centripetal acceleration must be equal to the gravitational acceleration ( g ):
[
g = frac{GM}{r^2}
]
where ( M ) is the mass of the star and ( G ) is the gravitational constant, approximately ( 6.674 times 10^{-11} , text{m}^3, text{kg}^{-1}, text{s}^{-2}