A space vehicle flies into a gravitational force field F(x, y, z)= y(b²x² + a²y²)i + x(b²x² + a²y²) j+czk with a flight path given by x(t) = a cost, y(t) = b sint, z(t) = c(4π - t) for a duration of 0 <≤ t ≤ 47 where a=2.8, b=2 and c=0.9. Compute the work done (following path integral): 4x F F. dr t=0
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- The radius Rh of a black hole is the radius of a mathematical sphere, called the event horizon, that is centered on the black hole. Information from events inside the event horizon cannot reach the outside world. According to Einstein's general theory of relativity, Rh = 2GM/c2, where M is the mass of the black hole and c is the speed of light. Suppose that you wish to study a black hole near it, at a radial distance of 48Rh. However, you do not want the difference in gravitational acceleration between your feet and your head to exceed 10 m/s2 when you are feet down (or head down) toward the black hole. (a) Take your height to be 1.5 m. What is the limit to the mass of the black hole you can tolerate at the given radial distance? Give the ratio of this mass to the mass MS of our Sun.Consider two particles: p at the origin (0,0,0) = R³ with mass M > 0, and q at the point/position vector 7 = (x, y, z) = R³ with mass m > 0. Let G be the universal gravitational constant. (We will assume the MKS system of units.) The force F = F (7) felt by the particle q due to its gravitational interaction with particle p is: GMm 7(7)= == 7, for all 7 = (x, y, z) € R³\{0} . 17 Also consider the function ƒ : R³\{♂} → R given by GMm f(x, y, z) := TT , for all 7 = (x, y, z) € R³\{0} . Fix an arbitrary point/position vector = (x, y, z) in R³\{♂}. 2, calculate the (3) Calculat cade of the vector (4) Calculate the direction of the vector ₹(7). (5) Assume that is the total force on the particle q. Calculate the instantaneous acceleration, d, of the particle q when it is at the point 7 = (x, y, z).An object is dropped from the top of a cliff 600 meters high. Its height above the ground t seconds after it is dropped is 600−4.9t^2. Determine its speed 9 seconds after it is dropped.
- When measuring with a muline, it was measured as n=10 when the speed was 1.1 m/s, and as n=32 when the speed was 1.5 m/s. What is the velocity formula?You plan to take a trip to the moon. Since you do not have a traditional spaceship with rockets, you will need to leave the earth with enough speed to make it to the moon. Some information that will help during this problem: mearth = 5.9742 x 1024 kgrearth = 6.3781 x 106 mmmoon = 7.36 x 1022 kgrmoon = 1.7374 x 106 mdearth to moon = 3.844 x 108 m (center to center)G = 6.67428 x 10-11 N-m2/kg2 1) On your first attempt you leave the surface of the earth at v = 5534 m/s. How far from the center of the earth will you get? 2) Since that is not far enough, you consult a friend who calculates (correctly) the minimum speed needed as vmin = 11068 m/s. If you leave the surface of the earth at this speed, how fast will you be moving at the surface of the moon? Hint carefully write out an expression for the potential and kinetic energy of the ship on the surface of earth, and on the surface of moon. Be sure to include the gravitational potential energy of the earth even when the ship is…Asap plzzxx
- The gravitational force exerted by the earth on a unit massat a distance r from the center of the planet is where M is the mass of the earth,R is its radius, and G isthe gravitational constant. Is F a continuous function of r ?A mass ?m is suspended from a massless spring of natural length 90 cm with the spring constant ?=10k=10 Nm-1 and causes the spring to extend by 9.5 cm. Assuming the gravitational field strength ?=9.8g=9.8 ms-2, calculate the value of the mass on the spring. Give your answer in SI units