Global warming is a cause for concern because even small changes in the Earth\'s

Question

Global warming is a cause for concern because even small changes in the Earth's temperature can have significant consequences. For example, if the Earth's polar ice caps were to melt entirely, the resulting additional water in the oceans would flood many coastal areas. Model the polar ice as having a combined mass of 2.65*1019 kg and forming two flat discs of radius 6.80*105 m. Calculate the resulting change in the duration of one day both in seconds and as a percentage.

? s
? %

Explanation / Answer

The change in the length of a day will be insignificant. This can be done by using conservation of angular momentum and the fact that the before and after the melting these values must be equal. Mass of each ice cap = m = 1.025x10^19 kg ... (combined mass of 2.05*10^19 kg) Moment of inertia of a cap = (1/2)m*R^2 ... R = radius of the earth Moment of inertia both caps = m*R^2 Moment of inertia of shell =(2/3)m*R^2 Moment of inertia of earth = IE = (2/5)M*R^2 ... assuming solid sphere of mass = M Angular momentum. w = angular velocity I = moment of inertia Angular momentum = L = I*w But w = V/R and P = 2*pi*R/V so that w = 2*pi/P So L = I*2*pi/P Angular momentum before must equal angular momentum after. Before: the moment of inertia will be that of the earth plus that of the two discs. Pb = period before the caps melt Lb = angular momentum before the caps melt Lb = [(2/5)M*R^2 + m*R^2]*2*pi/Pb After: the moment of inertia will be that of the earth plus that of the spherical water shell. Pa = period after the caps melt La = angular momentum after the caps melt La = [(2/5)M*R^2 + (2/3)m*R^2]*2*pi/Pa Equate Lb and La and get the ratio of Pa to Pb [(2/5)M*R^2 + (2/3)m*R^2] / [(2/5)M*R^2 + m*R^2] = Pa/Pb [1 + (5/3)m/M] / [1 + (5/2)m/M] = Pa/Pb... divide top and bottom by (2/5)M*R^2 Since M >> m the fractional change will be very small And since numerator is

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