Geological evidence based on several radiometric techniques has provided a scientifically well-accepted age for the Earth. Repre
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Answer:
a) w = 2.52 10⁷ rad / s, b) K / K₀ = 1.19 10⁴
Explanation:
a) We can solve this exercise using the conservation of angular momentum.
Initial instant. Before collapse
L₀ = I₀ w₀
Final moment. After the collapse
L_f = I w
angular momentum is conserved
L₀ = L_f
I₀ w₀ = I w (1)
The moment of inertia of a sphere is
I = 2/5 m r²
we take from the table the mass and diameter of the star
m = 1,991 10³⁰ kg
r₀ = 6.96 10⁸ m
r = 6.37 10⁶ m
to find the angular velocity let's use
w = L / T
where the length of a circle is
L = 2π r
T = 24 days (24 h / 1 day) (3600 s / 1h) = 2.0710⁶ s
we substitute
w = 2π r / T
wo = 2π 6.96 10⁸ / 2.07 10⁶
wo = 2.1126 10³ rad / s
we substitute in equation 1
w =
w = 2/5 mr₀² / 2/5 m r² w₀
w = () ² wo
w = (6.96 10⁸ / 6.37 10⁶) ² 2.1126 10³
w = 2.52 10⁷ rad / s
b) the kinetic energy ratio
K = ½ m w²
K₀ = ½ m w₀²
K = ½ m w²
K / K₀ = (w / wo) ²
K / K₀ = 2.52 10⁷ / 2.1126 10³
K / K₀ = 1.19 10⁴