Frodo Baggins has got himself in a pickle. He is carrying the One ring of power

Question

Frodo Baggins has got himself in a pickle. He is carrying the One ring of power toward the ominous Mount Doom, and the Lord of the Ring is causing Frodos dread to manifest as a physical force. This force makes it harder for frodo to move toward his destination in the following mathematical way:

Fdread = (111/x,-3y^3) = 111/x i hat -3y^3 j hat

Frodo prefers (,) notation but sauron prefers i j notation. The force is measured in Newtons, x is the horizontal distance between frodo and center of Mount Doom, y is the vertical distance Frodohas climbed off the flat ground.

a) The y-component of this force makes it so that the Ring feels heavier as Frodo climbs higher. At a height of 4m above the ground, how heavy will the Ring feel? (Neglect the actual weight of the Ring)

b) Since force is not constant here, use the calculus definition to calculate the work it does

Use this to compute the work done by the dread force as Frodo moves from xi - 17m to xf = 1m and yi = 0m to yf = 6m. Answer should be -1286J Use this number for part c

c) If Frodo's initial speed is 2.5m/s and Sam manages to do +2900J of work on poor frodo by carrying him up the mountain, what is frodos final speed? Frodo's mass is 35 kg (two works involved here - dont forget gravity - middle of earth = 9.8 m/s^2)

Explanation / Answer

a) 3*4^3 = 192 N

b) W = integral of F dx = 111 (ln(xfinal) - ln(xinitial)) - 3/4 (yf^4 - yi^4)


W = 111 * (-ln(17) + ln(1)) -3/4( (6^4-0^4))=-1286

c)

Ei + Wsam+ Wring = Ef

1/2*35*2.5^2 + 2900 - 1286 = 0.5*35*v^2 + 35*9.81*6

no solution for v
he doesnt make it up the mountain

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