Another possible energy source would be gravity. Imagine that the strange object contains a black hole of mass mbh. You drop a star of mass m into it. If the star of mass m starts at rest at infinity, what will the kinetic energy of the star be when it crosses the event horizon?
Give an equation for the energy of the star.
To determine the kinetic energy of the star when it crosses the event horizon of the black hole, we need to consider the conservation of energy.
In the situation described, the initial potential energy of the star at infinity will be converted into kinetic energy as it falls towards the black hole. At the event horizon, the star will still have some remaining kinetic energy.
The equation for the total energy (E) of the star can be expressed as the sum of its potential energy and kinetic energy:
E = K + U
Where:
K = Kinetic energy of the star
U = Potential energy of the star
Initially, when the star is at rest at infinity, its potential energy will be at its maximum value since it is far away from the gravitational influence of the black hole. As the star falls towards the black hole, its potential energy decreases and is converted into kinetic energy.
At the event horizon, the potential energy of the star will be completely converted into kinetic energy. Therefore, the equation for the energy of the star can be simplified to:
E = K
Hence, the kinetic energy of the star when it crosses the event horizon will be equal to its total energy at that point.
The equation for the energy of the star can be calculated using the principle of conservation of energy. The total energy of the star can be divided into its gravitational potential energy at infinity and its kinetic energy at the event horizon.
The gravitational potential energy at infinity can be given as:
PE_infinity = -G * (mbh * m) / (r_infinity)
where G is the gravitational constant, mbh is the mass of the black hole, m is the mass of the star, and r_infinity is the distance from the black hole to infinity (assuming it is very far away).
At the event horizon, the kinetic energy of the star is given by:
KE_event_horizon = 0.5 * m * v^2
where KE_event_horizon is the kinetic energy, m is the mass of the star, and v is the velocity of the star at the event horizon.
Since the star starts at rest at infinity, its initial velocity is zero, and therefore the kinetic energy at the event horizon will also be zero:
KE_event_horizon = 0
Thus, the kinetic energy of the star when it crosses the event horizon will be zero.