Question

1. the moon has a mass of 7.35 x 10^22 kg, a radius of 1,738 km, and an angular velocity of 1 revolution every 27.3 days. determine the moons angular momentum.

2.36 x 10^29 kg·m²/s
5.91 x 10^29 kg·m²/s
3.34 x 10^40 kg·m²/s
8.35 x 10^40 kg·m²/s

Explanation:

Step1: Convert days to seconds

1 day = 24 hours, 1 hour = 60 minutes, 1 minute = 60 seconds. So 27.3 days = 27.3×24×60×60 s = 2358720 s.

Step2: Calculate angular velocity $\omega$

The moon makes 1 revolution (which is $2\pi$ radians) in 27.3 days. So $\omega=\frac{2\pi}{2358720}\text{ rad/s}\approx 2.66\times 10^{-6}\text{ rad/s}$.

Step3: Calculate moment - of - inertia $I$ for a solid sphere

For a solid sphere, the moment - of - inertia formula is $I = \frac{2}{5}mr^{2}$. Given $m = 7.35\times 10^{22}\text{ kg}$ and $r=1.738\times 10^{6}\text{ m}$, then $I=\frac{2}{5}\times7.35\times 10^{22}\times(1.738\times 10^{6})^{2}$.
\[

$$\begin{align*} I&=\frac{2}{5}\times7.35\times 10^{22}\times 3.02\times10^{12}\\ &=8.86\times 10^{34}\text{ kg}\cdot\text{m}^{2} \end{align*}$$

\]

Step4: Calculate angular momentum $L$

The formula for angular momentum is $L = I\omega$. Substitute $I = 8.86\times 10^{34}\text{ kg}\cdot\text{m}^{2}$ and $\omega = 2.66\times 10^{-6}\text{ rad/s}$ into the formula.
\[

$$\begin{align*} L&=(8.86\times 10^{34})\times(2.66\times 10^{-6})\\ &=2.36\times 10^{29}\text{ kg}\cdot\text{m}^{2}/\text{s} \end{align*}$$

\]

Answer:

$2.36\times 10^{29}\text{ kg}\cdot\text{m}^{2}/\text{s}$