Question

The additional K.E. to be provided to a satellite of mass \(\mathrm{m}\) revolving around a planet of mass \(\mathrm{M}\), to transfer it from a circular orbit of radius \(\mathrm{R}_{1}\) to another radius \(\mathrm{R}_{2}\left(\mathrm{R}_{2}>\mathrm{R}_{1}\right)\) is (A) \(\operatorname{GMm}\left[\left(1 / R_{1}^{2}\right)-\left(1 / R_{2}^{2}\right)\right]\) \(\operatorname{GMm}\left[\left(1 / R_{1}\right)-\left(1 / R_{2}\right)\right]\) (C) \(2 \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]\) (D) \((1 / 2) \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]\)

Short answer

The additional kinetic energy needed to transfer a satellite from a circular orbit of radius $R_1$ to another radius $R_2 (R_2 > R_1)$ is: (D) $\frac{1}{2} GMm\left[\left(\frac{1}{R_1}\right)-\left(\frac{1}{R_2}\right)\right]$

Step-by-step solution

Find kinetic energy in the first (R1) orbit

To find the kinetic energy of the satellite in the orbit with radius R1, we can use the formula: \[T_1 = \frac{1}{2}mv_{1}^2\], where 'v1' is the initial orbital speed of the satellite in the R1 orbit. Since it is in a circular orbit, we can use the centripetal force, which is the gravitational force in this case: \[\frac{mv_{1}^{2}}{R_{1}} = \frac{GMm}{R_{1}^{2}}\].

Solve for 'v1'

Now, we need to solve for the initial orbital speed 'v1' using the centripetal force equation: \[v_{1}^2 = \frac{GM}{R_1}\].

Substitute 'v1' into the kinetic energy formula for the R1 orbit

We substitute the expression we got for v1 into the kinetic energy formula: \[T_1 = \frac{1}{2}m \cdot \frac{GM}{R_1}\].

Find kinetic energy in the second (R2) orbit

We repeat steps 1 to 3 for the R2 orbit. The kinetic energy of the satellite in the orbit with radius R2 is given by: \[T_2 = \frac{1}{2}mv_{2}^2\], So, \[\frac{mv_{2}^{2}}{R_{2}} = \frac{GMm}{R_{2}^{2}}\], \[v_{2}^2 = \frac{GM}{R_2}\], \[T_2 = \frac{1}{2}m \cdot \frac{GM}{R_2}\].

Calculate the difference in kinetic energy

To find the additional kinetic energy needed to transfer the satellite from the R1 orbit to the R2 orbit, we calculate the difference in kinetic energy: \[\Delta T = T_2 - T_1 = \frac{1}{2}m \cdot \frac{GM}{R_2} - \frac{1}{2}m \cdot \frac{GM}{R_1}\].

Simplify the expression

Now we simplify the expression for the difference in kinetic energy: \[\Delta T = \frac{1}{2} GMm\left(\frac{1}{R_2} - \frac{1}{R_1}\right)\]. Comparing the simplified expression with the choices given, we see that the additional kinetic energy needed to transfer the satellite from the R1 orbit to the R2 orbit is: (D) $(1 / 2) \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]$

Key concepts

Gravitational Force

Gravitational force is the attractive force that exists between any two masses. It is described by Sir Isaac Newton's law of universal gravitation. This force is given by the formula: \[F = \frac{GMm}{r^2}\]where:

• \(F\) is the gravitational force,
• \(G\) is the gravitational constant (\(6.674 \times 10^{-11} \ \text{Nm}^2/\text{kg}^2\)),
• \(M\) and \(m\) are the masses of the two objects,
• \(r\) is the distance between the centers of the two masses.

Gravitational force is a central force, meaning it acts along the line joining the centers of the two objects. It plays a crucial role in maintaining the orbits of planets and satellites. For satellites in orbit, this force acts as the centripetal force needed to keep them moving in circular paths. Without gravitational force, satellites would fly off into space.

Orbital Mechanics

Orbital mechanics is the study of the movements of objects in space, particularly those that are artificial, like satellites. It involves understanding how gravitational forces influence their trajectories.
In the case of a satellite orbiting a planet, its path forms an orbit, which can be circular or elliptical. The velocity and gravitational pull determine the shape and stability of this orbit.
Circular orbits, like those described in the exercise, imply that the satellite travels around the planet in a circle, with constant speed. The gravitational force provides the necessary centripetal force. The balance between these two forces allows the satellite to maintain its orbit without drifting away or falling towards the planet.
To transfer a satellite from one orbit to another (e.g., from \(R_1\) to \(R_2\)), we often use a transfer orbit, which changes the speed and kinetic energy of the satellite.

Energy Conservation

Energy conservation in orbital mechanics refers to the principle that the total mechanical energy of an object, like a satellite, remains constant in an isolated system without external forces. The total mechanical energy includes kinetic energy and potential energy.
Kinetic energy for a satellite in orbit is given by:\[T = \frac{1}{2}mv^2\].Potential energy due to gravity is described by:\[U = -\frac{GMm}{r}\].
The negative sign indicates that gravitational potential energy decreases with the distance from the planet.
When a satellite moves from one orbit to another, its kinetic and potential energies change. However, the sum of these energies remains constant if no external work is done on the system. This energy conservation principle helps calculate the additional kinetic energy needed for orbital changes.
Understanding energy conservation is essential for efficient satellite maneuvers and ensuring they reach their desired orbits.

Centripetal Force

Centripetal force is the force that keeps an object moving in a circular path. It acts towards the center of the circle. For satellites orbiting a planet, this centripetal force is actually provided by gravitational forces.
The formula for centripetal force is:\[F_c = \frac{mv^2}{r}\],where:

• \(F_c\) is the centripetal force,
• \(m\) is the mass of the object,
• \(v\) is the velocity of the object,
• \(r\) is the radius of the circular path.

In a stable orbit, the gravitational attraction must exactly match the centripetal force needed to keep the satellite in orbit. Therefore, \[\frac{mv^2}{r} = \frac{GMm}{r^2}\].
This balance ensures that the satellite maintains its velocity and orbit radius. If any of these factors change, adjustments must be made to maintain orbit. Understanding centripetal force is key to ensuring that satellites stay on course.