The Earth is 1.5 10 11 m from the Sun and takes a year to make one complete orbi

Question

The Earth is 1.5 1011 m from the Sun and takes a year to make one complete orbit. It rotates on its own axis once per day. It can be treated approximately as a uniform-density sphere of mass 6 1024 kg and radius 6.4 106 m (actually, its center has higher density than the rest of the planet, and the Earth bulges out a bit at the equator). Using this crude approximation, calculate the following.

(a) What is vCM?
m/s

(b) What is Ktrans?
J

(c) What is , the angular speed of rotation around its own axis?
rad/s

(d) What is Krot?
J

(e) What is Ktot?
J

Explanation / Answer

(A) w = 2pi /T

= 2 pi/ (365 x 24 x 3600s)

= 1.992 x 10^-7 rad/s

v = w r = (1.992 x 10^-7) (1.5 x 10^11m )

v = 3 x 10^4 m/s

(B) K trans = m v^2 /2

= (5.972 x 10^24) (3 x 10^4)^2 /2

= 2.67 x 10^33 J

(c) w = 2pi / T

= 2 pi / (24 x 3600)

= 7.27 x 10^-5 rad/s

(d) Krot = I w^2 /2

I = 2 m r^2 / 5

= 2 x 5.972 x 10^24 x (6.4 x 10^6)^2 / 5

= 9.78 x 10^37 kg m^2

Krot = I w^2 /2

= 2.585 x 10^29 J

(e) Ktot = Ktarns + Krot = 2.67 x 10^33 J

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