a) Calculate the angular momentum (in kg - m2/s) of Neptune in its orbit around the Sun. (The mass of Neptune is 1.020 x 1025 kg, the orbital radius is 4.497 × 10° km and the orbital period is 165 y.) 0.04063 ]× kg-m²/s b) Compare this angular momentum with the angular momentum of Neptune on its axis. (The radius of Neptune is 2.476 x 10 km and the rotation period is 16.11 h.) Lorbital Lrotation = 6138
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![### Angular Momentum of Neptune
#### (a) Calculate the Angular Momentum of Neptune in its Orbit around the Sun
To calculate the angular momentum (in \( \text{kg} \cdot \text{m}^2/\text{s} \)) of Neptune in its orbit around the Sun, we will use the given parameters:
- Mass of Neptune (\( m \)): \( 1.020 \times 10^{26} \) kg
- Orbital radius (\( r \)): \( 4.497 \times 10^9 \) km
- Orbital period (\( T \)): 165 years
The angular momentum (L) is given by:
\[
L = m \cdot v \cdot r
\]
where \( v \) is the orbital velocity, which can be calculated using:
\[
v = \frac{2 \pi r}{T}
\]
The calculated angular momentum is:
\[
0.04063 \ \text{kg} \cdot \text{m}^2/\text{s}
\]
#### (b) Compare This Angular Momentum with the Angular Momentum of Neptune on Its Axis
To compare this with the angular momentum of Neptune on its axis, we also need:
- Radius of Neptune (\( R \)): \( 2.476 \times 10^4 \) km
- Rotation period (\( T \)): 16.11 hours
The ratio of the orbital angular momentum (\( L_{\text{orbital}} \)) to the rotational angular momentum (\( L_{\text{rotation}} \)) is:
\[
\frac{L_{\text{orbital}}}{L_{\text{rotation}}} = 6138
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