Question

question 3
there are four planets with the masses and radii shown below. rank the gravitational field strength of the planets at their surface from greatest to least.
planet a: mass of \m\ and radius of
\
planet b: mass of \2m\ and radius of
\
planet c: mass of \m\ and radius of
/2\
planet d: mass of \2m\ and radius of \2r\
d > b > a > c
c > a > d > b
c > b > a > d
a > b > c = d

Explanation:

Step1: Recall the formula for gravitational field strength

The formula for gravitational field strength \(g\) at the surface of a planet is \(g=\frac{GM}{R^{2}}\), where \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(R\) is the radius of the planet.

Step2: Calculate \(g\) for each planet

• Planet A: \(g_{A}=\frac{GM}{R^{2}}\)
• Planet B: \(g_{B}=\frac{G(2M)}{R^{2}} = 2\frac{GM}{R^{2}}=2g_{A}\)
• Planet C: \(g_{C}=\frac{GM}{(R/2)^{2}}=\frac{GM}{\frac{R^{2}}{4}} = 4\frac{GM}{R^{2}}=4g_{A}\)
• Planet D: \(g_{D}=\frac{G(2M)}{(2R)^{2}}=\frac{2GM}{4R^{2}}=\frac{1}{2}\frac{GM}{R^{2}}=0.5g_{A}\)

Step3: Compare the values of \(g\)

We have \(g_{C} = 4g_{A}\), \(g_{B}=2g_{A}\), \(g_{A}=g_{A}\), \(g_{D}=0.5g_{A}\)

Answer:

C > B > A > D