Problem 90 - Master Resource Book in JEE Main Physics

Question

A comet of mass \(m\) moves in a highly elliptical orbit around the sun of mass \(M\). The maximum and minimum distances of the comet from the centre of the sun are \(r_{1}\) and \(r_{2}\) respectively. The magnitude of angular momentum of the comet with respect to the centre of sun is (a) \(\left[\frac{G M r_{1}}{\left(r_{1}+r_{2}\right)}\right]^{1 / 2}\) (b) \(\left[\frac{G M m r_{1}}{\left(r_{1}+r_{2}\right)}\right]^{1 / 2}\) (c) \(\left(\frac{2 G m^{2} r_{12}}{r_{1}+r_{2}}\right)^{1 / 2}\) (d) \(\left(\frac{2 G M m^{2} r_{12}}{\left(r_{1}+r_{2}\right)}\right)^{1 / 2}\)

Step-by-Step Solution

Answer

Option (b): \(\left[\frac{G M m r_{1}}{(r_{1}+r_{2})}\right]^{1 / 2}\).

Step 1: Use Conservation of Angular Momentum

The comet's movement in its orbit conserves angular momentum. Angular momentum \( L \) at the farthest point (\( r_2 \)) and nearest point (\( r_1 \)) to the sun can be expressed as: \( L = m v_1 r_1 = m v_2 r_2 \), where \( v_1 \) and \( v_2 \) are the velocities of the comet at \( r_1 \) and \( r_2 \), respectively.

Step 2: Apply Conservation of Energy

The total mechanical energy \( E \) at any point in the orbit is constant. The energy at the maximum distance \( r_1 \) is \( E = \frac{1}{2} m v_1^2 - \frac{G M m}{r_1} \) and at the minimum distance \( r_2 \) is \( E = \frac{1}{2} m v_2^2 - \frac{G M m}{r_2} \). Set these energies equal to each other: \( \frac{1}{2} m v_1^2 - \frac{G M m}{r_1} = \frac{1}{2} m v_2^2 - \frac{G M m}{r_2} \).

Step 3: Relate Velocity and Distance

From conservation of energy, express \( v_1^2 \) and \( v_2^2 \) in terms of \( r_1 \) and \( r_2 \):\[ v_1^2 = \frac{2 G M}{r_1} - \frac{2 G M}{r_1 + r_2} \]\[ v_2^2 = \frac{2 G M}{r_2} - \frac{2 G M}{r_1 + r_2} \].This gives us the relationship between velocities and distances from the sun.

Step 4: Solve for Angular Momentum

Using formulas from the previous steps, substitute \( v_1 \):\[ L = m r_1 \sqrt{\frac{2 G M}{r_1} - \frac{2 G M}{r_1 + r_2}} \].Simplifying, \( L \) becomes:\[ L = m \sqrt{\frac{2 G M r_1^2}{r_1 - r_2 + (r_1 + r_2)}} = m \sqrt{\frac{2 G M r_1^2}{2(r_1 + r_2)}} = \left(m \sqrt{\frac{G M r_1^2}{r_1 + r_2}}\right) \].

Step 5: Identify the Correct Option

Matching the derived expression for angular momentum, the correct option is (b):\[\boxed{\left[\frac{G M m r_{1}}{(r_{1}+r_{2})}\right]^{1 / 2}}\].

Key Concepts

Conservation of Angular Momentum

When discussing celestial bodies like comets, angular momentum plays a significant role in understanding their motion. Angular momentum is a conserved quantity, meaning that in the absence of external torques, it remains constant. For a comet orbiting the sun, this conservation principle ensures that the angular momentum at any two points in its orbit remains the same.
For example, if you consider the comet at its closest approach (perihelion, with distance \( r_1 \)) and its farthest point (aphelion, with distance \( r_2 \)), the comet's velocity and distance at each point are related through this conservation law. Mathematically, this can be expressed as:

• \( L = m v_1 r_1 = m v_2 r_2 \)

where \( v_1 \) and \( v_2 \) are the velocities at \( r_1 \) and \( r_2 \), respectively. This equation illustrates how the speed of the comet changes at different points in its orbit.
Understanding this principle can help predict the speeds of celestial objects at different points, providing insight into their orbital characteristics.

Conservation of Energy

Energy in celestial mechanics is another crucial concept for understanding cometary orbits. Like angular momentum, total mechanical energy in the system is conserved. This means that the energy of the comet remains constant along its path, exchanging between kinetic and potential energy but without net loss or gain.
The mechanical energy \(E\) at any position can be characterized by the equation:

• At a distance \( r_1 \) (aphelion): \( E = \frac{1}{2} m v_1^2 - \frac{G M m}{r_1} \)
• At a distance \( r_2 \) (perihelion): \( E = \frac{1}{2} m v_2^2 - \frac{G M m}{r_2} \)

Here, \( G \) is the gravitational constant and \( M \) is the mass of the sun. By equating these expressions, one can solve for the relationship between the velocities and distances:

• \( \frac{1}{2} m v_1^2 - \frac{G M m}{r_1} = \frac{1}{2} m v_2^2 - \frac{G M m}{r_2} \)

This formula allows us to understand how changes in distance affect the velocity and energy of the comet as it travels its elliptical orbit.

Mechanical Energy in Orbits

The concept of mechanical energy gives us deeper insights into orbits. Mechanical energy is a sum of potential and kinetic energy. In an elliptical orbit, like that of a comet, energy is continuously interchanged between these two forms.
When the comet is at its closest point to the sun, its speed—and hence its kinetic energy—is at its maximum due to conservation of energy. Conversely, at its farthest point, the speed and kinetic energy are reduced and potential energy is maximized.
This energy balance is a direct consequence of the gravitational forces acting on the comet, and understanding it is crucial for calculating velocities at various orbital points using the equation of energy conservation:

• The kinetic and potential energies adjust to ensure the total energy \( E \) remains constant.

This characteristic of maintaining constant mechanical energy helps us deduce important orbital elements and predict the behavior of the comet within its path.

Elliptical Orbits

Elliptical orbits are defined by their oval shapes, characterized by two focal points - one of which is occupied by a massive body like the sun in our solar system. These orbits are a staple in celestial mechanics and describe the paths taken by many comets and planets.
In these orbits, the distances from the orbiting body to the central mass vary as the body moves along the orbit. This variation results in changes in the velocity of the orbiting body, dictated by the laws of conservation of angular momentum and energy.
The shape of an elliptical orbit is described by its semi-major axis and eccentricity.

• The semi-major axis is half the longest diameter of the ellipse and affects the orbital period.
• The eccentricity, a measure of how "stretched" the ellipse is, affects how elongated the orbit is.

Understanding elliptical orbits helps us predict the positions and velocities of celestial objects and is essential for calculating trajectories in both natural and artificial orbital mechanisms.