Question

J.R.R. Tolkien wrote in his epic novel The Lord of the Rings that the One Ring “...did not seem always of the same size or weight; it shrank and expanded in an odd way...” For instance, in Peter Jackson’s film adaptation of The Fellowship of the Ring, Isildur cut the One Ring from the hand of the Dark Lord Sauron during the final battle on the slopes of Mount Doom, and once the ring was removed from the severed finger of Sauron, it contracted to the size of a large male finger, which is about 25 mm in diameter. Let’s assume the ring changes size according to thermal expansion and was made of a material with a coefficient of thermal expansion of 40 × 10−5 K−1. Suppose, too, that Sauron, who is often pictured to be standing in fire or lava, has a temperature of 1100◦ C. If the ring cools to a temperature of 30◦ C when it contracts, what is the diameter of Sauron’s finger if we assume the ring fit snuggly before it was removed?

Answer 1

According to thermal expansion,

when the ring is heated its circumference changes,

let us say its circumference changes from c₀ to c₁

let 0 represent sauron and 1 represent normal human

then,

(c₁ - c₀)/ c₀ = *dT

(d₁ - d₀)/ d₀ = *dT

(d₁ - d₀)/ d₀ = *dT

d₁ = d₀*(1+ *(T₁ - T₀))

25 = d₀*(1+ 40*10^-5(1100 - 30))

d₀ = 43.70 mm

this is the diameter of Sauron's finger