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Chapter 8: Q.16 (page 199)

A $9.4 脳 10^{21} k g$ moon orbits a distant planet in a circular orbit of radius $1.5 脳 10^{8} m$ . It experiences a $1.1 脳 10^{19} N$ gravitational pull from the planet. What is the moon鈥檚 orbital period in earth days?

Short Answer

Expert verified

The moon's orbital period in earth days is $= 26$ days

Step by step solution

01

Given information

Moon orbit's mass $= 9.4 脳 10^{21} k g$

Moon orbit's radius $= 1.5 脳 10^{8} m$

Gravitational pull $= 1.1 脳 10^{19} N$

02

Explanation

Gravitational pull $= F = \frac{G M m}{r^{2}}$

localid="1649397302533" $F = 1.1 脳 10^{19} N$

localid="1649397341555" $G = 6.67 脳 10^{- 11} N m^{2} k g^{- 2}$

$m = 9.4 脳 10^{21} kg$

localid="1649397362538" $r = 1.5 脳 10^{8} m$

Substitute the values

localid="1649397565238" $1.1 脳 10^{19} N = (\frac{(6.67 脳 10^{鈭� 11} N m^{2} k g^{- 2}) (9.4 脳 10^{21} k g)}{{(1.5 脳 10^{8} m)}^{2}})$

$鈬� M =$ Mass of the distant planet $= 3.947 脳 10^{23} kg$

Moon's orbital period in earth days is given by the expression,

$T = 2 蟺 \sqrt{\frac{r^{3}}{G (M + m)}}$

localid="1649397468079" $= 2 脳 3.14 \sqrt{(\frac{{(1.5 脳 10^{8} m)}^{3}}{(6.67 脳 10^{鈭� 11} N m^{2} k g^{- 2}) [(3.947 脳 10^{23} k g) + (9.4 脳 10^{21} k g)]})}$

$= 2248240 s$

localid="1649397499772" $= (\frac{2248240 s}{3600 s 脳 24 s}) days$

$= 26$ days

03

Final answer

The moon鈥檚 orbital period in earth days is

$= 26$ days

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