Passage
Black holes have played an important part in astrophysics for the last half-century. A black hole is a region of space where the force of gravity is especially strong and gravitational potential energy U is very large. Analogous to the electrostatic potential energy associated with electrical charges, U is defined as the work required to move an object of mass m from an infinite distance to a point at radial distance r within the gravitational field of a second mass M as given by
Equation 1where G is the gravitational constant. Near a black hole, the gravitational potential energy can become so negative that not even light has enough energy to escape the black hole's gravity.Escape velocity can be determined by launching an object of mass m with velocity v to an infinite radial distance from the surface of a central mass M (eg, a star or a planet) with radius R. For a black hole to exist, the escape velocity must be equal to the speed of light.Scientists have thoroughly studied the physics of black holes. Stars near the center of our galaxy have been observed to orbit an extremely massive, invisible object (millions of times more massive than our Sun) that is almost certainly a black hole. The orbital angular velocity ω of a star is defined as
ω=360°⋅fEquation 2where f is the frequency of rotation in hertz (Hz) . Likewise, black holes are not merely static masses but are often observed to rotate about their own axis. The interaction between the rotation of a black hole and matter falling into the hole is suspected to form powerful beams (jets) of ionized matter that project away from the black hole along its axis of rotation. Nevertheless, many aspects of how these jets are formed remains a mystery. Black holes will remain a fruitful area for research far into the future.
-Which of the following graphs best illustrates how the gravitational potential energy for a black hole varies with radial distance r ?

Question

Passage
Black holes have played an important part in astrophysics for the last half-century. A black hole is a region of space where the force of gravity is especially strong and gravitational potential energy U is very large. Analogous to the electrostatic potential energy associated with electrical charges, U is defined as the work required to move an object of mass m from an infinite distance to a point at radial distance r within the gravitational field of a second mass M as given by

Equation 1where G is the gravitational constant. Near a black hole, the gravitational potential energy can become so negative that not even light has enough energy to escape the black hole's gravity.Escape velocity can be determined by launching an object of mass m with velocity v to an infinite radial distance from the surface of a central mass M (eg, a star or a planet) with radius R. For a black hole to exist, the escape velocity must be equal to the speed of light.Scientists have thoroughly studied the physics of black holes. Stars near the center of our galaxy have been observed to orbit an extremely massive, invisible object (millions of times more massive than our Sun) that is almost certainly a black hole. The orbital angular velocity ω of a star is defined as

ω=360°⋅fEquation 2where f is the frequency of rotation in hertz (Hz) . Likewise, black holes are not merely static masses but are often observed to rotate about their own axis. The interaction between the rotation of a black hole and matter falling into the hole is suspected to form powerful beams (jets) of ionized matter that project away from the black hole along its axis of rotation. Nevertheless, many aspects of how these jets are formed remains a mystery. Black holes will remain a fruitful area for research far into the future.
-Which of the following graphs best illustrates how the gravitational potential energy for a black hole varies with radial distance r ?

Quizplus · Accepted Answer

11ecba2d_132c_2ea9_a3bf_0d32c4f3c0bf_MD0008_00 The total energy within a mechanical system is the sum of the system's kinetic and potential energy. Kinetic energy is derived from the motion of objects within a system whereas potential energy depends on the position of objects within the system. Because potential energy is related to position and not path, potential energy is associated with conservative forces (eg, gravitational, electrostatic, elastic forces). When establishing a reference point to be used for calculating potential energy, an infinite radial distance is chosen as a convenient location at which the potential energy is zero. Because potential energy measures the potential of conservative forces to do work, the change in potential energy (ΔU) relative to this point is equal and opposite to the work (W) done by conservative forces within the system: 11ecba2d_132c_2eaa_a3bf_6fe85673d35d_MD0008_00 Equation 1 in the passage shows that gravitational potential energy (U) is inversely proportional to the radial distance (r) separating the object mass (m) from the central mass (M) (eg, a black hole). The chosen reference point is an infinite radius where U is zero and the work done by gravity in moving an object from infinity closer to the central mass is positive. Therefore, ΔU is negative and the potential energy at a finite radius (U_r) is always negative: 11ecba2d_132c_2eab_a3bf_0916e19be75f_MD0008_00 11ecba2d_132c_2eac_a3bf_5d3e76f9100e_MD0008_00 11ecba2d_132c_2ead_a3bf_636fcec5154e_MD0008_00 Because U_r is always negative, the curve of inverse proportionality is located within the bottom-right quadrant of an xy-coordinate system. Therefore, the graph that best fits the trend of Equation 1 is an inversely proportional plot located below the x-axis (Choice B). (Choice A) This graph depicts an inverse relationship between U and r. However, Equation 1 includes a negative sign on the right side of the equation, meaning that U is always negative rather than positive. (Choices C and D) These graphs depict large (negative or positive) potential energy at large radial distances and zero potential energy at r = 0. Therefore, both are inconsistent with an inversely proportional relationship. Educational objective: Gravitational potential energy measures the potential of gravity to do work on a mass. Conventionally, gravitational potential energy is expressed using negative numbers and is inversely proportional to the radial distance between masses.

Passage Black Holes Have Played an Important Part in Astrophysics for for the Last