In elliptical orbit of a planet

Options

A. angular momentum about centre of sun is constant
B. potential energy is constant
C. kinetic energy is constant
D. total mechanical energy is constant

Answer

To solve the question regarding the properties of a planet in an elliptical orbit around the Sun, we will analyze each option step by step.

### Step 1: Analyze Angular Momentum
- **Concept**: Angular momentum (L) of a planet in orbit is given by the formula \( L = mvr \), where \( m \) is the mass of the planet, \( v \) is its velocity, and \( r \) is the distance from the Sun.
- **Observation**: In an elliptical orbit, the net torque acting on the planet is zero because the gravitational force acts along the line connecting the planet and the Sun. Therefore, the angular momentum remains constant.
- **Conclusion**: The first option, "angular momentum about the center of the Sun is constant," is **correct**.

### Step 2: Analyze Potential Energy
- **Concept**: The gravitational potential energy (U) between two masses is given by \( U = -\frac{G m_1 m_2}{r} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between them.
- **Observation**: As the planet moves in its elliptical orbit, the distance \( r \) between the planet and the Sun changes. Since potential energy is inversely proportional to \( r \), it varies as the distance changes.
- **Conclusion**: The second option, "potential energy is constant," is **incorrect**.

### Step 3: Analyze Kinetic Energy
- **Concept**: The kinetic energy (K) of the planet is given by \( K = \frac{1}{2} mv^2 \).
- **Observation**: In an elliptical orbit, the speed of the planet varies; it moves faster when closer to the Sun (periapsis) and slower when farther away (apoapsis). Thus, the kinetic energy also changes as the speed changes.
- **Conclusion**: The third option, "kinetic energy is constant," is **incorrect**.

### Step 4: Analyze Total Mechanical Energy
- **Concept**: The total mechanical energy (E) of the system is the sum of kinetic and potential energy, \( E = K + U \).
- **Observation**: In an isolated system (like the planet-Sun system), the total mechanical energy remains constant over time, even though kinetic and potential energy may change individually.
- **Conclusion**: The fourth option, "total mechanical energy is constant," is **correct**.

### Final Summary of Correct Options
- The correct options are:
- A: Angular momentum about the center of the Sun is constant (Correct)
- B: Potential energy is constant (Incorrect)
- C: Kinetic energy is constant (Incorrect)
- D: Total mechanical energy is constant (Correct)