In planetary motion, the angular Momentum conservation leads to the law of

Options

A. orbits
B. areas
C. periods
D. conservation of kinetic energy

Answer

To solve the question regarding the conservation of angular momentum in planetary motion and its relation to Kepler's laws, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Angular Momentum in Planetary Motion**:
- In planetary motion, the angular momentum \( L \) of a planet about the Sun is given by the formula:
\[ L = mvr \]
where \( m \) is the mass of the planet, \( v \) is its tangential velocity, and \( r \) is the distance from the Sun.

2. **Conservation of Angular Momentum**:
- According to the law of conservation of angular momentum, if no external torque acts on the system, the angular momentum remains constant. This means:
\[ L = mvr = \text{constant} \]
- As a planet moves in its elliptical orbit, its distance \( r \) from the Sun changes, which affects its velocity \( v \).

3. **Relating Angular Momentum to Kepler's Laws**:
- Kepler's Second Law, also known as the Law of Areas, states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This can be expressed mathematically as:
\[ \frac{dA}{dt} = \text{constant} \]
- Here, \( dA \) is the area swept out in a small time interval \( dt \).

4. **Deriving the Law of Areas**:
- The area swept out in a time interval \( dt \) can be related to the angular momentum. The area swept out is proportional to the angular momentum because:
\[ dA = \frac{1}{2} r^2 d\theta \]
- Since \( L \) is constant, the rate at which area is swept out must also be constant, leading to Kepler's Second Law.

5. **Conclusion**:
- Therefore, the conservation of angular momentum in planetary motion leads to Kepler's Second Law, which is the Law of Areas.

### Final Answer:
In planetary motion, the conservation of angular momentum leads to the **Law of Areas** (Kepler's Second Law).

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