Let's analyze the motion of an object under the gravitational pull of another object. This situation is governed by Newton's law of universal gravitation, which describes a central force.

Understanding Gravitational Motion

When an object moves under the influence of gravity from another object (like a planet orbiting a star), the force is always directed towards the center of the second object. This force is known as a central force.

Conserved Quantities Analysis

We need to check which quantities remain constant during this motion:

• Linear Momentum:

  Linear momentum is defined as \(\vec{p} = m\vec{v}\), where \(m\) is the mass and \(\vec{v}\) is the velocity. For linear momentum to be conserved, the net force acting on the object must be zero (\(\sum \vec{F} = 0\)). In this case, there is a significant gravitational force acting on the object. This force causes the object's velocity vector (\(\vec{v}\)) to change in direction and often magnitude, meaning the linear momentum (\(\vec{p}\)) is generally not conserved.

• Angular Momentum:

  Angular momentum is defined as \(\vec{L} = \vec{r} \times \vec{p}\), where \(\vec{r}\) is the position vector from the center of the gravitational pull. The torque (\(\vec{\tau}\)) acting on the object is given by \(\vec{\tau} = \vec{r} \times \vec{F}\). Since the gravitational force (\(\vec{F}\)) is a central force, it is always parallel to the position vector (\(\vec{r}\)). The cross product of two parallel vectors is zero. Therefore, the torque is zero (\(\vec{\tau} = 0\)). According to the conservation of angular momentum, if the net torque is zero, angular momentum is conserved. Thus, angular momentum is conserved in this scenario.

• Mechanical Energy:

  Mechanical energy (\(E\)) is the sum of kinetic energy (\(K\)) and potential energy (\(U\)). The gravitational force is a conservative force. For conservative forces, the work done by the force depends only on the initial and final positions, not the path taken. When only conservative forces do work, the total mechanical energy (\(E = K + U\)) of the system remains constant. Kinetic energy (\(K = \frac{1}{2}mv^2\)) changes as the object's speed changes (it increases when closer to the source of gravity and decreases when farther away), and potential energy (\(U = -G\frac{m_1m_2}{r}\)) also changes. However, their sum, the mechanical energy, remains constant. Thus, mechanical energy is conserved.

Conclusion on Conservation

Based on the analysis:

• Linear momentum is not conserved.
• Angular momentum is conserved.
• Mechanical energy (Kinetic Energy + Potential Energy) is conserved.

Therefore, the quantities that remain conserved are angular momentum and mechanical energy.

Final Answer Selection

Comparing this with the options provided:

• Option 1: Only linear momentum (Incorrect)
• Option 2: Potential energy and linear momentum (Incorrect)
• Option 3: Kinetic energy and linear momentum (Incorrect)
• Option 4: Angular momentum and mechanical energy (Correct)

The correct option is the one stating that both angular momentum and mechanical energy are conserved.