Questions: Unit 2: Forces Exam Free Response Questions 1. A spacecraft of mass m is in a clockwise circular orbit of radius R around Earth, as shown in the figure. The mass of Earth is ME. Note: Figure not drawn to scale. a. On the figure to the right, draw and label the forces (not components) that act on the spacecraft. Each force must be represented by a distinct arrow starting on, and pointing away from, the spacecraft. b. i. Derive an equation for the orbital period T of the spacecraft in terms of m, M E, R, and physical constants, as appropriate. If you need to draw anything other than what you have shown in part (a) to assist Note: Figure not drawn to scale. in your solution, use the space below. Do NOT add anything to the figure in part (a). ii. A second spacecraft of mass 2m is placed in a circular orbit with the same radius R. Is the orbital period of the second spacecraft greater than, less than, or equal to the orbital period of the first spacecraft? Greater than Less than Equal to Briefly explain your reasoning.

Unit 2: Forces Exam Free Response Questions 1. A spacecraft of mass m is in a clockwise circular orbit of radius R around Earth, as shown in the figure. The mass of Earth is ME. Note: Figure not drawn to scale. a. On the figure to the right, draw and label the forces (not components) that act on the spacecraft. Each force must be represented by a distinct arrow starting on, and pointing away from, the spacecraft. b. i. Derive an equation for the orbital period T of the spacecraft in terms of m, M E, R, and physical constants, as appropriate. If you need to draw anything other than what you have shown in part (a) to assist Note: Figure not drawn to scale. in your solution, use the space below. Do NOT add anything to the figure in part (a). ii. A second spacecraft of mass 2m is placed in a circular orbit with the same radius R. Is the orbital period of the second spacecraft greater than, less than, or equal to the orbital period of the first spacecraft? Greater than Less than Equal to Briefly explain your reasoning.
Transcript text: Unit 2: Forces Exam Free Response Questions 1. A spacecraft of mass $m$ is in a clockwise circular orbit of radius $R$ around Earth, as shown in the figure. The mass of Earth is $M_{E}$. Note: Figure not drawn to scale. a. On the figure to the right, draw and label the forces (not components) that act on the spacecraft. Each force must be represented by a distinct arrow starting on, and pointing away from, the spacecraft. b. i. Derive an equation for the orbital period $T$ of the spacecraft in terms of $m, M E, R$, and physical constants, as appropriate. If you need to draw anything other than what you have shown in part (a) to assist Note: Figure not drawn to scale. in your solution, use the space below. Do NOT add anything to the figure in part (a). ii. A second spacecraft of mass $2 m$ is placed in a circular orbit with the same radius R. Is the orbital period of the second spacecraft greater than, less than, or equal to the orbital period of the first spacecraft? $\qquad$ Greater than $\qquad$ Less than $\qquad$ Equal to Briefly explain your reasoning.
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Solution

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Solution Steps

Step 1: Identify the forces acting on the spacecraft.

The only force acting on the spacecraft is the gravitational force exerted by the Earth. This force is directed towards the center of the Earth.

Step 2: Draw the force vector.

Draw an arrow pointing from the spacecraft towards the center of the Earth and label it $F_g$.

Step 3: Derive an equation for the orbital period.

The gravitational force provides the centripetal force required for circular motion. Thus, we can equate the gravitational force to the centripetal force:

\(F_g = F_c\)

\(\frac{GM_Em}{R^2} = \frac{mv^2}{R}\)

where \(G\) is the gravitational constant and \(v\) is the orbital speed of the spacecraft.

The orbital speed can be related to the orbital period \(T\) and radius \(R\) as follows:

\(v = \frac{2\pi R}{T}\)

Substituting this expression for \(v\) into the previous equation and solving for \(T\), we get:

\(\frac{GM_Em}{R^2} = \frac{m(\frac{2\pi R}{T})^2}{R}\)

\(\frac{GM_E}{R^2} = \frac{4\pi^2 R}{T^2}\)

\(T^2 = \frac{4\pi^2 R^3}{GM_E}\)

\(T = \sqrt{\frac{4\pi^2 R^3}{GM_E}}\)

\(T = 2\pi\sqrt{\frac{R^3}{GM_E}}\)

Step 4: Determine the relationship between the orbital periods of the two spacecraft.

The orbital period derived in the previous step does not depend on the mass of the spacecraft. Therefore, the orbital period of the second spacecraft (mass \(2m\)) is equal to the orbital period of the first spacecraft (mass \(m\)).

Final Answer

a. The force vector points from the spacecraft towards the center of the Earth, labeled as \(F_g\). b. i. \\(\boxed{T = 2\pi\sqrt{\frac{R^3}{GM_E}}}\\) ii. Equal to. The orbital period does not depend on the mass of the spacecraft.

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