This set of Orbital Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Equations of Motion & Conservation”.

1. While deriving the equation of motion, what are the other forces and torques assumed to be?

a) Zero

b) Infinite

c) 1

d) Negligible

View Answer

Explanation: When the equation of motion is derived for a to-body problem, the bodies experience gravitational force only. All the other forces and torques are assumed to be zero.

2. If the mass of two bodies are m_{1} and m_{2} respective with position vectors as r_{1} and r_{2}. Then what is the position vector of center of mass?

a) r_{c} = \(\frac{m_1 r_1 – m_2 r_2}{m_1 – m_2}\)

b) r_{c} = \(\frac{(m_1 + m_2)(r_1 + r_2)}{m_1 m_2}\)

c) r_{c} = \(\frac{m_1 r_1 + m_2 r_2}{m_1 + m_2}\)

d) r_{c} = \(\frac{m_1 r_1 + m_2 r_2}{m_1 m_2}\)

View Answer

Explanation: If there is an inertial system X,Y,Z assumed having two point masses m

_{1}and m

_{2}, with position vectors as r

_{1}and r

_{2}, then the position vector of center of mass is calculated using the formula:

r

_{c}= \(\frac{m_1 r_1 + m_2 r_2}{m_1 + m_2}\) Where, r

_{c}is the position vector of center of mass.

3. Which of these is not an assumption considered while deriving the equation of motion?

a) The bodies ae considered to be point masses

b) The point masses never touch

c) The center of mass of two bodies does not lie in the center

d) No external force acts on the bodies apart from gravitational force

View Answer

Explanation: While deriving the equation of motion, there are three main assumptions made as follows:

• Bodies of mass m1 and m2 are spherical point masses.

• The point masses never touch.

• No external force other than gravitational force acts on the two bodies.

4. If satellite of mass 10,000 kg orbits around Earth of mass 5.98 × 10^{24} kg at a distance of 100 million km, then when is the gravitational potential energy force between the two masses?

a) 39.9 × 10^{6} J/kg

b) 33.1 × 10^{6} J/kg

c) 48.9 × 10^{6} J/kg

d) 53.2 × 10^{6} J/kg

View Answer

Explanation: m

_{1}= 10,000 kg, m

_{2}= 5.98 × 10

^{24}kg, r = 100 × 10

^{9}km

The potential energy of the gravitational force is given by:

V = –\(\frac{Gm_1 m_2}{r}\)

Substituting the values, we get:

V = –\(\frac{6.673 × 10^{-11} × 10,000 × 5.98 × 10^{24}}{100 × 10^9}\) = 39.9 × 10

^{6}J/kg

5. Center of mass of a two-body system is non-accelerating.

a) True

b) False

View Answer

Explanation: In a system with two point masses m

_{1}, m

_{2}having mutual gravitational force acting on each other, the center of mass lies somewhere along the line joining the two masses. The center of mass of this system is non accelerating having a constant velocity.

6. How is the acceleration affected if surface area of the spherical satellite orbiting the Earth becomes twice?

a) Becomes half

b) Becomes 1/4^{th}

c) Becomes twice

d) Become four times

View Answer

Explanation: Using the differential equation of motion for the two body system relation we know that-

Where, is the acceleration

Therefore,

From the above relation we obtain that acceleration of the body is inversely proportional to the square of area of surface of sphere. Thus, if the surface area of satellite becomes twice, then the acceleration becomes 1/4

^{th}.

7. Angular momentum is conserved in a two-body equation.

a) True

b) False

View Answer

Explanation: The angular momentum of the system is given by: \(\vec h = \vec r × \vec v\). On differentiating the term we get:

\(\frac{d\vec h}{dt} = \frac{d}{dt}(\vec r × \frac{d\vec r}{dt})\) = 0

Thus the angular momentum vector is constant and is conserved in two-body motion.

8. On what parameter does the Energy of the orbit not vary?

a) Eccentricity

b) True anomaly

c) Inclination

d) Radius

View Answer

Explanation: Energy of the orbit is given by the formula:

ε = \(\frac{v^2}{2} – \frac{\mu}{r} = -\frac{\mu}{2a}\)

Thus, energy is a constant in all the positions of the orbit and is independent of the eccentricity e. It is only dependent on the size of the orbit and not the shape of the orbit.

9. What happens to the maximum potential energy of the satellite if the distance between the satellite and the attracting body tends to infinity?

a) Tends to zero

b) Remains constant

c) Tends to infinity

d) Increases proportionally

View Answer

Explanation: The satellite while orbiting an attracting body has negative potential energy and its maximum potential energy tends to zero when the radius tends to infinity.

10. What is the value of torque in a two-body system?

a) τ = \(\vec r\) × m

b) τ = \(\vec r × \vec r\)

c) τ = \(\vec r × \vec v\)

d) Zero

View Answer

Explanation: When there is a two-body system, there is an interaction and gravitational force acts parallel to the displacement vector. The torque is given by:

τ = \(\vec r × \vec F\)

Where torque on the particle is due to the force \(\vec F\). But since the force vector acts parallel to the displacement vector, torque becomes zero and thus the angular momentum remains constant.

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