If an object collapses to a black hole, [problem 9-99 for student then the escape velocity at the surface of the black hole is equal to the speed of light. Determine the radius of the black hole, if the mass is 4.1 times the mass of the sun.
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- Calculate the gravitational constant g, in SI units, for alocation on the surface of the sun.There is a small sphere of mass m at a height H, from the center of a uniform ring of radius R and mass M. The center of the ring is placed at the origin of a Cartesian coordinate system. The x and y directions line in the plane of the ring, while the z-direction is positive upwards.’ Take the x-axis to be directed towards the right (in the plane of the ring). Write an expression for the x-component of the gravitational force Fx on the mass m, where G is the gravitational constant.A mood of a mass m and raduis a is orbiting a planet of mass M and if radius b at a distance d (center to center) in a circular orbit. Derive an expression for the speed of the mood v in terms of M,d and the gravitational constant G.
- when we calculate escape speeds, we usually do so with the assumption that the object from which we are calculating escape speed is isolated. This is, of course, generally not true in the solar system. Show that the escape speed at a point near a system that consists of two stationary massive spherical objects is equal to the square root of the sum of the squares of the escape speeds from each of the two objects considered individually.A)Artificial satellites are used to monitor weather conditions on Earth, forsurveillance and for communications. Such satellites may be placed in ageo-synchronous orbit in a low polar orbit Describe the properties of the geo-synchronous (or geo-stationary) orbit and the advantages it offers when a satellite is used for communications B)A satellite of mass m travels at angular speed coin a circular orbit ataheight h above the surface of a planet of mass M and radius R. b i)Using these symbols, give an equation that relates the gravitationalforce on the satellite to the centripetal force. b ii)Use your equation from part (b)(i) to show that the orbital period,T, of the satellite is given by : T^2=(4π^2)/GM * (R/h)^3 b iii)Explain why the period of a satellite in orbit around the Earthcannot be less than 85 minutes. Your answer should include a calculationto justify this value. Mass of the Earth = 6.4 x 10^24 kg and the radius of theEarth = 6.4 x 10^6 m.C) Describe and explain what…Space debris left from old satellites and their launchers is becoming a hazard to other satellites. (a) Calculate the speed of a satellite in an orbit 900 km above Earth's surface. (b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite's orbit at an angle of 90°. What is the velocity of the rivet relative to the satellite just before striking it? (c) If its mass is 0.500 g, and it comes to rest inside the satellite, how much energy in joules is generated by the collision? (Assume the satellite's velocity does not change appreciably, because its mass is much greater than the rivet's.)
- 10-13. Refer to Example 10.3 concerning the deflection from the plumb line of a particle falling in Earth's gravitational field. Take g to be defined at ground level and use the zeroth order result for the time-of-fall, T= V2h/g. Perform a calculation in second approximation (i.e., retain terms in w?) and calculate the southerly deflec- tion. There are three components to consider: (a) Coriolis force to second order (C), (b) variation of centrifugal force with height (C2), and (c) variation of gravi- tational force with height (Cs). Show that each of these components gives a result equal to h2 o sin A cos A with C 2/3, C2 5/6, and Cs = 5/2. The total southerly deflection is therefore (4h-w² sin A cos A)/g. toThe free-fall acceleration on the surface of Jupiter is about two and one half times that on the surface of the Earth. The radius of Jupiter is about 11.0 RE (RE= Earth's radius = 6.4 x 106 m). Find the ratio of their average densities, Pjupiter/PEarth-