Q: Given the enthalpy of formation data for CH$_3$OH, calculate the total interatomic equilibrium potential energy for the molecule.

$U = \sum \text{Enthalpy of formation of atoms} - \text{Enthalpy of formation of molecule}$

Question 1

Two identical pendulums of length $l = 0.4$ m are connected by a light coupling spring. If $g = 9.8$ m/s$^2$ and the period of one pendulum (with the other clamped) is found to be 1.25 s, what is the period of the other normal mode when both pendulums are free to oscillate?

Accepted Answer

$T = 2\pi \sqrt{l/g + k/m}$

Question 2

A plant is hung from a spring ($k = 8$ N/m) and oscillates with a frequency of 2 Hz and amplitude of 12 cm. What is the acceleration of the plant when its velocity is half the maximum velocity?

Accepted Answer

$a = \omega^2 \sqrt{A^2 - (v_{max}/2\omega)^2}$

Question 3

A cantilever beam of length $l = 20$ mm and mass $m_e = 0.1$ mg has a spring constant $k = \frac{3EI}{l^3}$ and is actuated by an external force $F_0 \cos \omega t$. What is the expression for the resonant angular frequency $\omega_0$ of the oscillator?

Accepted Answer

$\omega_0 = \sqrt{\frac{k}{m_e}}$

Question 4

For a driven oscillator, the displacement can be expressed in terms of absorptive ($A_{abs}$) and dispersive ($A_{dis}$) amplitude components. Which component does not contribute to the average power absorbed?

Accepted Answer

Dispersive component $A_{dis}$

Question 5

Four passengers with a combined mass of 250 kg compress the spring of a car by 4.00 cm. If the loaded car has a period of vibration of 1.08 s, what is the period of vibration of the empty car?

Accepted Answer

$T_{empty} = T_{loaded} \sqrt{\frac{m_{loaded}}{m_{empty}}}$

Question 6

For heat transfer in fluids, which molecular process is fundamentally similar between conduction and convection, and which is significantly different?

Accepted Answer

Both involve molecular collisions, but convection includes bulk movement

Question 7

Given the enthalpy of formation data for CH$_3$OH, calculate the total interatomic equilibrium potential energy for the molecule.

Accepted Answer

$U = \sum 	ext{Enthalpy of formation of atoms} - 	ext{Enthalpy of formation of molecule}$

Question 8

What structure would you expect Potassium Iodide (KI) to show at ambient conditions of 298 K and 1 atm?

Accepted Answer

Ionic crystalline lattice

Question 9

Given the electronic structures: Si $1s^2 2s^2 2p^6 3s^2 3p^2$ and Au $1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10} 4s^2 4p^6 4d^{10} 4f^{14} 5s^2 5p^6 5d^{10} 6s^1$, which material would be the better conductor of heat when solid?

Accepted Answer

Gold, due to its free electrons

Question 10

What is the significance of the Fermi level in defining the thermal and electrical properties of solids?

Accepted Answer

It is the highest occupied energy level at absolute zero and governs electron behavior

Question 11

A heat engine consists of two isothermal segments at temperatures $T_1$ and $T_2$ ($T_2 > T_1$), connected by two isobaric segments. What is the efficiency of the engine expressed in terms of $T_1$ and $T_2$ only?

Accepted Answer

$\eta = 1 - \frac{T_1}{T_2}$

Question 12

Consider internal energy $U$ and an isochoric, reversible process. Show that $C_v = T \left( \frac{\partial S}{\partial T} \right)_V$. What does $C_v$ represent?

Accepted Answer

Heat capacity at constant volume

Question 13

Show that the ratio of heat capacity at constant pressure and volume is equal to the ratio of isothermal and adiabatic compressibility: $\frac{C_p}{C_v} = \frac{K_T}{K_S}$. What does this ratio physically represent?

Accepted Answer

The relative ability of a substance to change volume under different thermodynamic conditions

Question 14

A wire carries current $I$ and two paths (A and B) are drawn around it. Along which path is the line integral of the magnetic field greater?

Accepted Answer

Path A, if it encloses more current

Question 15

Using nodal analysis, what is the general approach to determine the voltage $V_x$ in a circuit with multiple resistors and sources?

Accepted Answer

Set up node equations using Kirchhoff's Current Law and solve for node voltages

Question 16

A circuit contains a $160/45^\circ$ V source, a 3 $\Omega$ resistor, and a 1 $\Omega$ inductor. What is the total impedance seen by the voltage source?

Accepted Answer

$Z = 3 + j1 \, \Omega$

Question 17

A long straight wire carries a current $i(t) = a + bt$ near a stationary rectangular loop. What is the induced emf in the loop for $0 < t < 2.0$ s?

Accepted Answer

$\mathcal{E} = -\frac{d\Phi}{dt}$

Question 18

Which physical constant is used to relate the energy of a photon to its frequency?

Accepted Answer

Planck's constant $h$

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