Question

Planet Z is $10,000km$in diameter. The free-fall acceleration on Planet Z is role="math" localid="1648089747827" $8.0m/s^2.$

(a) What is the mass of Planet Z?

(b) What is the free-fall acceleration$10,000km$above Planet Z’s north pole?

Short answer

(a) The mass of planet Z is$1.25\times {10}^{25}kg.$

(b) The free-fall acceleration $10000km$above Planet Z’s north pole is $2m/s^2.$

Step-by-step solution

Given information (a)

Diameter of the planet = $10000km$, free-fall acceleration = $8m/s^2.$

Calculation (a)

The formula for free-fall acceleration is given by :$g=\frac{GM}{R^2}.$

Substituting the given values, in equation :

role="math" localid="1648091089811" $$\begin{gathered} \Rightarrow \left(8.0\mathrm{m}/{\mathrm{s}}^2\right)=\frac{\left(6.67\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2\right)M}{{\left(1.0\times {10}^7\mathrm{m}\right)}^2}. \\ \Rightarrow M=\frac{\left(8.0\mathrm{m}/{\mathrm{s}}^2\right){\left(1.0\times {10}^7\mathrm{m}\right)}^2}{\left(6.67\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2\right)} \\ =1.2\times {10}^{25}\mathrm{kg}. \end{gathered}$$

Final answer (a)

The mass of planet Z is $1.25\times {10}^{25}kg.$

Given information (b)

Height from the north pole $h=10000\mathrm{km}.$

Calculation (b)

The formula for free-fall acceleration at a height h is given by :$g^{\text{'}}=\frac{GM}{(R+h{)}^2}.$

Height given in this case is $h=R$, hence $h=R$is substituted to the above equation :

$$\begin{gathered} g^{\text{'}}=\frac{GM}{(R+R{)}^2} \\ =\frac{GM}{4R^2} \\ =\frac{1}{4}\frac{GM}{R^2} \\ =\frac{1}{4}g \\ =\frac{1}{4}\left(8.0\mathrm{m}/{\mathrm{s}}^2\right) \\ =2.0\mathrm{m}/{\mathrm{s}}^2. \end{gathered}$$

Final answer (b)

The free-fall acceleration $10000km$above Planet Z’s north pole is$2m/s^2.$