Question #204153

The moon Titan orbits the planet Saturn with a period of 1.9 × 106 s. The average radius of

this orbit is 1.6 × 109 m.

a) What is Titan’s centripetal acceleration?

b) What is Titans Velocity?


1
Expert's answer
2021-06-09T08:05:06-0400

The information given is

The period of a complete orbit is T=1.9×106  sT=1.9\times 10^{6}\;\text{s}

The average radius of the orbit is. r=1.6×109  sr=1.6\times 10^{9}\;\text{s}


Considering the circular orbit.


The angular speed is given by

w=2  π  radTw=\dfrac{2\;\pi\;\text{rad}}{T}


Where.

2  π  rad2\;\pi\;\text{rad} is the angular displacement of a complete orbit

TT is the period of a complete orbit.


Evaluating numerically.

w=2  π  radTw=2  π  rad1.9×106  sw=3.3×106rad/sw=\dfrac{2\;\pi\;\text{rad}}{T}\\ w=\dfrac{2\;\pi\;\text{rad}}{1.9\times 10^{6}\;\text{s} }\\ w= 3.3\times 10^{-6} \text{rad}/\text{s}



The centripetal acceleration is given by.

ac=w2  ra_{c}=w^{2}\;r


Where.

ww is the angular speed

rr is the radius.


Evaluating numerically.

ac=w2  rac=(3.3×106rad/s)2×1.6×109  sac=1.7×102  m/s2a_{c}=w^{2}\;r\\ a_{c}=( 3.3\times 10^{-6} \text{rad}/\text{s} )^{2}\times 1.6\times 10^{9}\;\text{s} \\ a_{c}=1.7\times 10^{-2}\;\text{m}/\text{s}^{2}


Answer A

The centripetal acceleration is ac=1.7×102  m/s2\displaystyle \color{red}{\boxed{a_{c}=1.7\times 10^{-2}\;\text{m}/\text{s}^{2}}}


Part b


The linear velocity is given by

V=wrV=w\cdot r


Where


ww is the angular speed

rr is the radio


Evaluating numerically

V=wrV=3.3×106rad/s×1.6×109  sV=5.3×103  m/sV=w\cdot r\\ V= 3.3\times 10^{-6} \text{rad}/\text{s}\times 1.6\times 10^{9}\;\text{s}\\ V=5.3\times 10^{3}\;\text{m}/\text{s}


Answer B

The velocity is V=5.3×103  m/s\displaystyle \color{red}{\boxed{V=5.3\times 10^{3}\;\text{m}/\text{s}}}


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