Subfield: Mechanics

- Proton mass, $m_{p}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Neutron mass, $m_{n}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Electron mass, $m_{e}=9.11 \cdot 10^{-31} \mathrm{~kg}$
- Avogadro's constant, $N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1}$
- Universal gas constant, $R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$
- Boltzmann's constant, $k_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K}$
- Electron charge magnitude, $e=1.60 \cdot 10^{-19} \mathrm{C}$
- 1 electron volt, $1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J}$
- Speed of light, $c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}$
- Universal Gravitational constant,

$$
G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2}
$$

- Solar Mass

$$
M_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg}
$$

- Acceleration due to gravity, $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$
- 1 unified atomic mass unit,

$$
1 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2}
$$

- Planck's constant,

$$
h=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s}
$$

- Permittivity of free space,

$$
\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right)
$$

- Coulomb's law constant,

$$
k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2}
$$

- Permeability of free space,

$$
\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}
$$

- Magnetic constant,

$$
\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A}
$$

- 1 atmospheric pressure,

$$
1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa}
$$

- Wien's displacement constant, $b=2.9$. $10^{-3} \mathrm{~m} \cdot \mathrm{K}$
- Stefan-Boltzmann constant,

$$
\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}
$$

Question: A rocket with mass of 563.17 (not including the mass of fuel) metric tons sits on the launchpad of the Kennedy Space Center (latitude $28^{\circ} 31^{\prime} 27^{\prime \prime} \mathrm{N}$, longitude $80^{\circ} 39^{\prime} 03^{\prime \prime} \mathrm{W}$ ), pointing directly upwards. Two solid fuel boosters, each with a mass of $68415 \mathrm{~kg}$ and providing $3421 \mathrm{kN}$ of thrust are pointed directly downwards.
The rocket also has a liquid fuel engine, that can be throttled to produce different amounts of thrust and gimbaled to point in various directions. What is the minimum amount of thrust, in $\mathrm{kN}$, that this engine needs to provide for the rocket to lift vertically (to accelerate directly upwards) off the launchpad?

Assume $G=6.674 \times 10^{-11} \frac{\mathrm{N} \cdot \mathrm{m}^{2}}{\mathrm{~s}^{3}}$, and that the Earth is a perfect sphere of radius $6370 \mathrm{~km}$ and mass $5.972 \times 10^{24} \mathrm{~kg}$ that completes one revolution every $86164 \mathrm{~s}$ and that the rocket is negligibly small compared to the Earth. Ignore buoyancy forces.

Solution:
Note the additional information provided in the problem (latitude, Earth radius, revolution period), which makes it clear that the effect of the rotation of the Earth must also be considered.
We first compute local gravitational acceleration:

$$
g=\frac{G M}{R^{2}}=9.823 \frac{\mathrm{m}}{\mathrm{s}^{2}}
$$



And also acceleration due to the Earth's rotation:

$$
a=(R \cos \theta) \omega^{2}=0.02976 \frac{\mathrm{m}}{\mathrm{s}^{2}}
$$

Then vertical acceleration is:

$$
g-a \cos \theta=9.7965 \frac{\mathrm{m}}{\mathrm{s}^{2}}
$$

And horizontal acceleration:

$$
a \sin \theta=0.0142 \frac{m}{s^{2}}
$$

The total mass of the craft is 700 metric tons, so the needed forces for liftoff are:

$$
\left(F_{x}, F_{y}\right)=(6857527,9948) \mathrm{N}
$$

Subtracting out the effect of the solid fuel boosters, the force the liquid fuel engine needs to provide is:

$$
\left(F_{x}, F_{y}\right)=(15527,9948) \mathrm{N}
$$

Final answer: $18.44$ kN