Question

question #6
a space agency sends a probe through space to a planet larger than earth. the larger planet has the
same mass as earth, but is twice the diameter.
how would the weight of the probe on the larger planet compare to its weight on earth?
a it would be one - fourth its weight on earth.
b it would be half of its weight on earth.
c it would be twice its weight on earth.
d it would be four times its weight on earth.

Explanation:

Step1: Recall the formula for gravitational force

The weight of an object on a planet is given by \(F = \frac{GMm}{R^{2}}\), where \(G\) is the gravitational constant, \(M\) is the mass of the planet, \(m\) is the mass of the probe, and \(R\) is the radius of the planet.

Step2: Analyze the radii of the two planets

Let the radius of Earth be \(R_{E}=d/2\). The radius of the larger planet \(R_{L}=(2d)/2 = d\).

Step3: Calculate the weight on Earth (\(F_E\)) and on the larger planet (\(F_L\))

For Earth: \(F_E=\frac{GMm}{(d/2)^{2}}=\frac{4GMm}{d^{2}}\)
For the larger planet: \(F_L=\frac{GMm}{d^{2}}\)

Step4: Find the ratio \(\frac{F_L}{F_E}\)

\(\frac{F_L}{F_E}=\frac{\frac{GMm}{d^{2}}}{\frac{4GMm}{d^{2}}}=\frac{1}{4}\)

Answer:

A. It would be one - fourth its weight on Earth.