Question

1. Steve, with mass m, is ready to squat jump Kale’s insecurities—represented by n moles of a solid with total mass M—out of Earth’s atmosphere. He decides to model his squat as a spring with constant k compressing vertically down a distance d from equilibrium, and the jump as releasing that spring, launching himself and Kale’s insecurities to a maximum height h above equilibrium. He does everything perfectly.

a) When he gets to that maximum height, he finds that Kale’s insecurities have heated by ∆T. How much energy went into heating them ?.

b) Assuming Steve’s temperature did not change and no other interactions were present, how far did Steve compress his model spring?

Answer 1

Atoms in solids have 3 degrees of freedom

Internal energy of solids E_in = 3nRT

when the temperature of Kale's has raised by T

change in internal energy = 3nR T - energy taken to heat Kale's

R = 8.31 J/mole/K - gas const.

b) Steve and Kale have gone to a height h

change in PE = (m+M)gh

if the spring is compressed by d , PE stored in the spring = kd²/2

Total spring energy is used to raise them by a height h and heating up Kale's, there are no other interactions. Conserving the total energy

kd² = (m+M)gh + 3nR T

compression of spring

d = { ( (m+M)gh + 3nR T )/k }^1/2