Question

It was a bright and breezy day in New York, the air was a cool 20 degrees Celsius, when Tony Stark, a.k.a the Iron Man's day took a dramatic turn as he got news of Mandarin's attack in the windy city of Chicago, and decided to immediately fly over there, in his iron body-armor, whose cavity had enough space to fit a man of 4,984 $c{m}^3$ in volume, but not more.

Being a smart man, he did a quick check of two important things - his own body volume, which he found was currently 4.980 $c{m}^3$, and the temperature in Chicago, which was a cold 8 degrees Celsius that day. Knowing that the coefficient of volume expansion for iron, $\gamma$, is $33.3\times {10}^{-6}$${/}^{\circ }C$, he decided to go. Check if that was a smart decision, by finding out whether the new volume of the iron suit when he reaches Chicago will crush him or not.

• A. (he is going to be alright)
• B. (it's going to be an uncomfortable encounter)
• C. (he will have to take his suit off and call the Hulk)
• D. (perfect fit!).

Answer 1

The correct option is A (he is going to be alright)

Mr. Stark's suit is made of iron, which should contract upon cooling with a coefficient $\gamma$ = $33.3\times {10}^{-6}$${/}^{\circ }C$. As he flies from New York to Chicago, there occurs a temperature drop of ${12}^{\circ }C$.

$\therefore$$\Delta T$ = $T_{Chicago}$ - $T_{NY}$ = ($8^{\circ }C$ - ${20}^{\circ }C$) = $-{12}^{\circ }C$

The volume, $V_{NY}$, of the suit's cavity in New York = $4,984c{m}^3$.

$\therefore$ The volume of the cavity in Chicago will be -

$V_{Chicago}$ = $V_{NY}(1+\gamma \Delta T)$

= $4.984[1+33.3\times {10}^{-6}\times (-12\left)\right]c{m}^3$

$\approx$ $4.982c{m}^3$.

Awesome! The suit will still fit Tony Stark comfortably in Chicago, so he can just concentrate on Mandarin's onslaught. I feel much less worried about Chicago now!