A very massive star with an M mass has a geometric shape of a sphere hollow, where the mass is distributed on the surface only with the radius R. Determine the gravitational potential inside (r < R) and outside (r > R), where r is radial distance from the center of the star. (Hint: You can use Gauss's Law to gravitational field).
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distributed on the surface only with the radius R.
Determine the gravitational potential inside (r < R) and outside (r > R), where r is radial distance from
the center of the star. (Hint: You can use Gauss's Law to gravitational field)."
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- a) Derive the escape speed vII (sometimes called the second cosmic speed) from the surface of body of radius R and mass M, using energy conservation. Assume that the planet does not move or spin, so that the total mechanical energy E of a test particle both on the surface and at infinity is equal to zero, and the speed at infinity is zero. Use the formula come up in the last part, Compute this escape speed (in km/s) from Venus, Mars, Jupiter and Neptune. b) Evaluate vIII from the heliocentric orbit of the 4 planets (Venus, Mars, Jupiter and Neptune) mentioned above, assuming circularity of their orbits. Compare the computed speeds with the escape speeds from their surfaces (vII). Notice a big difference in the situation around the inner and the outer planets. What is it due to? A fly-by’s of small bodies near a planet can result in the small particle being accelerated to a final speed close to vII w.r.t. the planet. Draw conclusions as to which planets are able to accelerate small…(a) What is the escape speed on a spherical asteroid whose radius is 502 km and whose gravitational acceleration at the surface is 0.870 m/s? (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 647 m/s? (c) With what speed will an object hit the asteroid if it is dropped from 665.5 km above the surface? (a) Number i Units (b) Number Units (c) Number UnitsNothing can escape the event horizon of a black hole, not even light. You can think of the event horizon as being the distance from a black hole at which the escape speed is the speed of light, 3.00 ×× 1088 m/sm/s, making all escape impossible. What is the radius of the event horizon for a black hole with a mass 7.5 times the mass of the sun? This distance is called the Schwarzschild radius.?
- PlzPRO BLEM 10: A PARTICLE IN A PARABOLIC BOWL := kr Suppose a particle of mass m lives inside a parabolic bowl. The height z of the bowl is given, as a function of it's radius as z = kr², where k is a number. On a particular day the particle finds itself at a height zo as measured from the bottom of the parabola. At that instant it has a horizontal speed vo. Ignoring friction calculate (1) for a given value of horizontal velocity, that is we will let vo → Vh, the particle will move in a perfect circle. What is this v,? Answer this part of the question by drawing a free body diagram and applying New ton's laws. (2) Now, suppose vo > vh, where, vh is the value you calculated above. What is the maximum height reached by the particle? Answer this part of the question by evoking the conservation of energy, and remember to discuss what happens to angular momentum here. (3) Now, suppose vo = 0, but the particle is still at 20. Let us assume that 20 is small and that he particle will undergo…Zero, a hypothetical planet, has a mass of 4.5 x 1023 kg, a radius of 3.2 x 106 m, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial kinetic energy of 5.0 x 107 J, what will be its kinetic energy when it is 4.0 x 106 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?
- (a) What is the escape speed on a spherical asteroid whose radius is 274 km and whose gravitational acceleration at the surface is 0.444 m/s2? (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 311 m/s? (c) With what speed will an object hit the asteroid if it is dropped from 289.4 km above the surface?(a) Calculate how much work is required to launch a spacecraft of mass m from the surface of the earth (mass mE, radius RE) and place it in a circular low earth orbit—that is, an orbit whose altitude above the earth’s surface is much less than RE. (As an example, the International Space Station is in low earth orbit at an altitude of about 400 km, much less than RE = 6370 km.) Ignore the kinetic energy that the spacecraft has on the ground due to the earth’s rotation. (b) Calculate the minimum amount of additional work required to move the spacecraft from low earth orbit to a very great distance from the earth. Ignore the gravitational effects of the sun, the moon, and the other planets. (c) Justify the statement “In terms of energy, low earth orbit is halfway to the edge of the universe.”(a) Evaluate the gravitational potential energy (in J) between two 6.00 kg spherical steel balls separated by a center-to-center distance of 19.0 cm. (b) Assuming that they are both initially at rest relative to each other in deep space, use conservation of energy to find how fast (in m/s) will they each be traveling upon impact. Each sphere has a radius of 5.20 cm. m/s
- (a) Evaluate the gravitational potential energy (in J) between two 4.00 kg spherical steel balls separated by a center-to-center distance of 19.0 cm. (b) Assuming that they are both initially at rest relative to each other in deep space, use conservation of energy to find how fast (in m/s) will they each be traveling upon impact. Each sphere has a radius of 5.50 cm. m/sIn introductory physics laboratories, a typical Cavendish balance for measuring the gravitational constant G uses lead spheres with masses of 1.20 kg and 12.0 g whose centers are separated by about 4.90 cm. Calculate the gravitational force between these spheres, treating each as a particle located at the center of the sphere. Need Help? Master It Read ItAssume a planet is a uniform sphere of radius R that (somehow) has a narrow radial tunnel through its center. Also assume we can position an apple anywhere along the tunnel or outside the sphere. Let Fp be the magnitude of the gravitational force on the apple when it is located at the planet's surface. How far from the surface (what multiple of R) is there a point where the magnitude of the gravitational force on the apple is 0.5 FR if we move the apple (a) away from the planet and (b) into the tunnel? (a) Number: Units: (b) Number: Units: