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)
Cm is revolving around sun
In 1 year, it completes 1 orbit
distance covered , d = 2*pi*r
d = 2*pi*(1.5*10^11 + 6.4*10^6)
= 9.425*10^11 m

time taken,
t = 1 year
=3.154*10^7 s

Vcm = distance covered / time taken
= (9.425*10^11)/(3.154*10^7)
= 2.99*10^4 m/s

b)
Ktrans = 0.5*m*Vcm^2
= 0.5*(6*10^24)*(2.99*10^4)^2
= 2.68*10^33 J

c)
time taken to rotate = 1 day = 86400 s
angle through which it rorate in 1 rotation = 2*pi radians
w = 2*pi/86400
= 7.27*10^-5 rad/s

d)
Moment of inertia of solid sphere, I = (2/5)*m*R^2
= (2/5)*(6*10^24)*(6.4*10^6)^2
= 9.83*10^37 Kg.m^2
K rot = 0.5*I*w^2
= 0.5 *(9.83*10^37)*(7.27*10^-5 )^2
= 2.6*10^29 J

e)
K tot = K rot + K trans =2.6*10^29 + 2.68*10^33 J = 2.68*10^33 J

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