$$\begin{array} { | l | c | c | c | c | } \hline & \text { Mass } & \text { Radius } & \text { Orbital radius } & \text { Orbital period } \ \hline \text { Moon A } & 4.0 \times 10 ^ { 20 } \mathrm {~kg} & \text { unknown } & 2.0 \times 10 ^ { 8 } \mathrm {~m} & 4.0 \times 10 ^ { 6 } \mathrm {~s} \ \hline \text { Moon B } & 1.5 \times 10 ^ { 2 } 0 \mathrm {~kg} & 2.0 \times 10 ^ { 5 } \mathrm {~m} & 3.0 \times 10 ^ { 8 } \mathrm {~m} & \text { unknown } \ \hline \end{array}$$ Mithra is an unknown planet that has two airless moons,A and B,in circular orbits around it.The table summarizes the hypothetical data about these moons.If you dropped a laser at the surface of Moon B,at what rate would it accelerate toward the ground? (G = 6.67 × 10^-11 N ∙ m²/kg²) A)0.10 m/s² B)0.15 m/s² C)0.20 m/s² D)0.25 m/s² E)0.30 m/s²

Question

Quizplus · Accepted Answer

We can use Newton's law of gravitation to determine the acceleration due to gravity experienced by a laser at the surface of Moon B. 
From the table, we know that Moon B has a mass of $1.5 	imes 10^{20}$ kg and a radius of $2.0 	imes 10^5$ m. The distance between the center of Moon B and its orbital radius is given as $3.0 	imes 10^8$ m. 
Using Newton's law of gravitation, we have:
$$F = G \frac{mM}{r^2}$$
where $F$ is the force of gravity, $G$ is the gravitational constant, $m$ is the mass of the object (in this case, the laser), $M$ is the mass of Moon B, and $r$ is the radius of Moon B. 
Solving for $F$, we get:
$$F = G \frac{mM}{r^2} = (6.67 	imes 10^{-11} \mathrm{~N \cdot m^2/kg^2})(\frac{m}{4 	imes 10^{20} \mathrm{~kg}})(1.5 	imes 10^{20} \mathrm{~kg}) (\frac{1}{2 	imes 10^5 \mathrm{~m}})^2 = 1.25 	imes 10^{-6} \mathrm{~N}$$
The force of gravity is equal to the weight of the laser, so we can use $F = ma$ to find the acceleration $a$:
$$a = \frac{F}{m} = \frac{1.25 	imes 10^{-6} \mathrm{~N}}{m}$$
Since we want the acceleration in terms of meters per second squared, we need to convert the mass of the laser from kilograms to grams:
$$a = \frac{1.25 	imes 10^{-6} \mathrm{~N}}{m} = \frac{1.25 	imes 10^{-3} \mathrm{~N}}{1000m} = 1.25 	imes 10^{-6} \mathrm{~m/s^2}$$
Therefore, the laser would accelerate toward the ground at a rate of 0.25 m/s$^2$. The best choice is D.

Mithra Is an Unknown Planet That Has Two Airless Moons,A