3. (35 points; 5 points each) In this problem, you will estimate the change in t

Question

3. (35 points; 5 points each) In this problem, you will estimate the change in the rotation rate of a sphere if a small body on its surface changes its latitude. Although this is a gross simplification, it can be used to estimate the order of magnitude change in the rotation rate of the Earth if due to global warming the polar ice caps or Greenland ice sheet fully melted and the melted water redistributed predominantly closer to the equator. Or it can be used if a major earthquake causes a land mass near the surface of the earth to significantly shiftThis happened during the Japan earthquake in 2011 where that entire country permanently moved by 8 feet! As shown in Figure 4, consider a large sphere of mass M and radius R with a small particle of mass m on its surface at initial latitude above the sphere's equator. The sphere and particle are joined to each other and rotating at an initial angular speed about the z axis. Due to internal forces and torques, the particle moves to a new latitude thus changing the mass distnbution of the sphere particle system. Because of conservation of angular momentum, the sphere particle system then rotating at a new angular velocity as s own in Figure 5. a) What is the initial moment of inertia lpi of the particle about the z axis in terms of m, R, and ? b) What is the total initial moment of inertial of the sphere-particle system about the z-axis if the moment of inertia of the sphere is lsphere = 2 MR2 ? Express your answer in terms of m, R-4 and M and w? c What is the total initial angular momentum L of the sphere-particle system in terms of m, M, R. d) What is the final moment of inertia , of the particle about thez axis in terms of m, R, and ? e) What is the total final moment of inertia I of the sphere-particle system about the z-axis if the moment of inertia of the sphere is lspher.-2 MR2 ? Express your answer in terms of m. R. and M f) What is the total final angular momentum L of the sphere-particle system in terms of m. M. R. and ? g) Using the law of conservation of angular momentum and your equations from c) and f derive an equation for the change in angular velocity A-a,- about the z axis in terms of all variables except and R z-axis 77n Figure 4. Sphere of mass M and radius R with a small particle of mass m on its surface at initial latitude both rotating at initial angular speed Figure 5. Sphere of Figure 4 after the small particle moves to a new latitude and now both rotating at angular speed about the z axis about the z axis.

Explanation / Answer

a) horizontal distance of the mass m from the z axis is, r=Rcos(i)

Ipi=mr2 = m*R2*cos2(i)

b) Total initial moment of inertia, Ii=Isphere+Ipi

Ii= (2/5)MR2 + m*R2*cos2(i)

c) Total initial angular momentum, Li = Ii*wi

Li= ((2/5)MR2 + m*R2*cos2(i))*wi

d) here, r= Rcos()

Ip = mr2 = m*R2*cos2()

e) I= (2/5)MR2 + m*R2*cos2()

f) L=I*w

L= ((2/5)MR2 + m*R2*cos2())*w

g) w=wi*(((2/5)MR2 + m*R2*cos2(i))/((2/5)MR2 + m*R2*cos2()))

w=wi*Li / L

w = w-wi = wi*Li/L - wi

= wi*((Li-L) / L)

=wi*((cos2(i)-cos2()) / ((2M/5m)+cos2()))

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