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baldbrain
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Homework Statement
Let g be the acceleration due to gravity at the Earth's surface and K be the rotational kinetic energy of the Earth. Suppose the Earth's radius decreases by 2%. Keeping all other quantities constant,
(a) g increases by 2% and K increases by 2%
(b) g increases by 4% and K increases by 4%
(c) g decreases by 4% and K decreases by 2%
(d) g decreases by 2% and K decreases by 4%
Homework Equations
g=GM/R2 , where M & R is the mass and the radius of the Earth respectively
K=(1/2)Iω2 , where I is the moment of inertia of the Earth about its axis of rotation and ω is it's angular velocity about the same axis
The Attempt at a Solution
(dR/R)100 = 2% ...(decrease)
Since all other quantities are constant,
i) g=GM/R2 ⇔ g ∝ R-2
⇒ dg/g = 2(dR/R)
⇒ (dg/g)100 = 2((dR/R)100)
= 2(2) = 4% ....(increase)
Since there is inverse proportionality, g increases by 4%
ii) K=(1/2)Iω2
Now, assuming the Earth to be a homogeneous sphere of uniform mass density, its moment of inertia about the diameter is
I=(2/5)MR2
Therefore K= (1/2)(2/5)MR2ω2 = (1/5)MR2ω2
Keeping all other quantities constant,
K ∝ R2
⇒dK/K = 2(dR/R)
⇒(dK/K)100 = 2((dR/R)100)
= 2(2) = 4% .... (decrease)
Since there is direct proportionality, K decreases by 4%Hence, g increases by 4% and K decreases by 4%
So, I think option (b) should've been - g increases by 4% and K decreases by 4%, instead of K increases by 4%
Thoughts?
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