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•Let's say the Sun is located at the origin of the (x, y) plane. The evil, but powerful, super-villain Galactus uses his energy absorption powers to drain al the planets and the asteroid belt of all their energy, which means they stop al of their movement around the Sun. Worse, once they've stopped, they go from being motionless to being sucked straight towards the Sun due to gravity, as was discussed two homeworks ago. Recall from class that the gravitational force exerted from the Sun on the planets has a magnitude of GMm, where m is the mass of the object being sucked towards the Sun, and r is the distance from the object being sucked to the origin. Assume that we have units where GM = 5, so the magnitude of the force is 54. Assume that when Galactus stops the planets and asteroids, Venus in located at (x, y) = (3,0), Earth is at (x, y) = (3,4), and the asteroid belt has an almost circular, but actually elliptic, path about the Sun described by r(t) = 6.02 cos(t)i+ 5.98 sin(t)j where t e [0, 27). Further m = 10 for Venus, m = 11 for Earth, and the density for the asteroid belt is given by p(x, y) = 3.01x² + 2.99y². Finally, assume that all objects are incinerated by the Sun's heat when they get one unit away from the origin. (That is, they are incinerated as soon as they hit the circle x² + y² = 1.)

•Let's say the Sun is located at the origin of the (x, y) plane. The evil, but powerful, super-villain Galactus uses his energy absorption powers to drain al the planets and the asteroid belt of all their energy, which means they stop al of their movement around the Sun. Worse, once they've stopped, they go from being motionless to being sucked straight towards the Sun due to gravity, as was discussed two homeworks ago. Recall from class that the gravitational force exerted from the Sun on the planets has a magnitude of GMm, where m is the mass of the object being sucked towards the Sun, and r is the distance from the object being sucked to the origin. Assume that we have units where GM = 5, so the magnitude of the force is 54. Assume that when Galactus stops the planets and asteroids, Venus in located at (x, y) = (3,0), Earth is at (x, y) = (3,4), and the asteroid belt has an almost circular, but actually elliptic, path about the Sun described by r(t) = 6.02 cos(t)i+ 5.98 sin(t)j where t e [0, 27). Further m = 10 for Venus, m = 11 for Earth, and the density for the asteroid belt is given by p(x, y) = 3.01x² + 2.99y². Finally, assume that all objects are incinerated by the Sun's heat when they get one unit away from the origin. (That is, they are incinerated as soon as they hit the circle x² + y² = 1.)

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•Let's say the Sun is located at the origin of the (r, y) plane. The evil, but powerful, super-villain Galactus uses his energy absorption powers to drain all the planets and the asteroid belt of all their energy, which means they stop all of their movement around the Sun. Worse, once they've stopped, they go from being motionless to being sucked straight towards the Sun due to gravity, as was discussed two homeworks ago. Recall from class that the gravitational force exerted from the Sun on the planets has a magnitude of GMm, where m is the mass of the object being sucked towards the Sun, and r is the distance from the object being sucked to the origin. Assume that we have units where GM = 5, so the magnitude of the force is 54. Assume that when Galactus stops the planets and asteroids, Venus in located at (x, y) = (3,0), Earth is at (x, y) = (3,4), and the asteroid belt has an almost circular, but actually elliptic, path about the Sun described by r(t) = 6.02 cos(t)i+ 5.98 sin(t)j where te [0, 27). Further m = 10 for Venus, m = 11 for Earth, and the density for the asteroid belt is given by p(x, y) = 3.01x? + 2.99y². Finally, assume that all objects are incinerated by the Sun's heat when they get one unit away from the origin. (That is, they are incinerated as soon as they hit the circle r2 + y? = 1.) d) If we approximate the asteroid path by the circle r(t) = 6 cos(t)i + 6 sin(t)j where t e [0, 27) and we approximate the density by p(x, y) = 3x² + 3y², determine the work done bv the Sun to incinerate the asteroid belt.
Transcribed Image Text:•Let's say the Sun is located at the origin of the (r, y) plane. The evil, but powerful, super-villain Galactus uses his energy absorption powers to drain all the planets and the asteroid belt of all their energy, which means they stop all of their movement around the Sun. Worse, once they've stopped, they go from being motionless to being sucked straight towards the Sun due to gravity, as was discussed two homeworks ago. Recall from class that the gravitational force exerted from the Sun on the planets has a magnitude of GMm, where m is the mass of the object being sucked towards the Sun, and r is the distance from the object being sucked to the origin. Assume that we have units where GM = 5, so the magnitude of the force is 54. Assume that when Galactus stops the planets and asteroids, Venus in located at (x, y) = (3,0), Earth is at (x, y) = (3,4), and the asteroid belt has an almost circular, but actually elliptic, path about the Sun described by r(t) = 6.02 cos(t)i+ 5.98 sin(t)j where te [0, 27). Further m = 10 for Venus, m = 11 for Earth, and the density for the asteroid belt is given by p(x, y) = 3.01x? + 2.99y². Finally, assume that all objects are incinerated by the Sun's heat when they get one unit away from the origin. (That is, they are incinerated as soon as they hit the circle r2 + y? = 1.) d) If we approximate the asteroid path by the circle r(t) = 6 cos(t)i + 6 sin(t)j where t e [0, 27) and we approximate the density by p(x, y) = 3x² + 3y², determine the work done bv the Sun to incinerate the asteroid belt.
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