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.
-A star orbiting a black hole in a clockwise direction at a radial distance of 1.0 × 10⁶ km is acted upon by a counterclockwise friction force of 250 N perpendicular to the orbit radius. What is the torque on the star?

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.
-A star orbiting a black hole in a clockwise direction at a radial distance of 1.0 × 10⁶ km is acted upon by a counterclockwise friction force of 250 N perpendicular to the orbit radius. What is the torque on the star?

Quizplus · Accepted Answer

11ecba2d_132e_78ae_a3bf_ab9706b5d82f_MD0008_00 A torque (τ) is a rotational force caused by a translational force (F) applied at some radial distance (r) from a center of rotation. Torques can cause rotation about a central axis or oppose any rotation already present. The magnitude of a torque is defined as: 11ecba2d_132e_9fbf_a3bf_dbf34e68f6f0_MD0008_00 where θ is the angle between the translational force and the radial vectors. When r and F are perpendicular (θ = 90°), the torque is maximal: 11ecba2d_132e_9fc0_a3bf_55f073f1f705_MD0008_00 11ecba2d_132e_9fc1_a3bf_7bcbf9d3d03e_MD0008_00 Conversely, when r and F are parallel (θ = 0°), the magnitude of torque is zero (the force has no rotational influence). In this question, a star orbits a black hole in a clockwise direction at a radial distance of 1.0 × 10⁶ km. The translational force applied on the star acts in a counterclockwise direction (opposite to the clockwise orbit of the star), perpendicular to the radial vector. Consequently, θ is 90° and the magnitude of the torque is: 11ecba2d_132e_9fc2_a3bf_818bffb39f62_MD0008_00 11ecba2d_132e_9fc3_a3bf_a9d9d1465307_MD0008_00 11ecba2d_132e_c6d4_a3bf_e3d28aa48e8d_MD0008_00 Therefore, the star is subjected to a counterclockwise torque with a magnitude of 2.5 × 10¹¹ N∙m (Choice D). Educational objective: Torques are forces that influence the rotational motion of a system in a clockwise or counterclockwise direction. The magnitude of a torque is the product of the applied translational force, the radial distance from the center of rotation, and the sine of the angle between the translational force and radial vectors.

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