Introduction to Classical Mechanics · Problem 9.36

Angular momentum, Part II (General L) — Morin

Physics — Mechanics · quadrupole

Problem

Consider an arbitrarily shaped body of mass mm whose CM is at the origin. Using the law of cosines, the gravitational potential of a mass MM at position R\mathbf{R} is V(R)=−∫GM dmR2+r2−2Rrcos⁡β,(9.98)V(\mathbf{R}) = -\int \frac{GM\,dm}{\sqrt{R^2 + r^2 - 2Rr\cos\beta}}, \tag{9.98} where the integration runs over the volume of the body, and β\beta is the angle that the position vector r\mathbf{r} of an arbitrary point in the body makes with the vector R\mathbf{R}. ((a)) Assuming that all points in the body satisfy r≪Rr \ll R, show that an approximate expression for the potential is V(R)≈−GMmR−GM2R3∫r2(3cos⁡2β−1) dm,(9.99)V(\mathbf{R}) \approx -\frac{GMm}{R} - \frac{GM}{2R^3}\int r^2(3\cos^2\beta - 1)\,dm, \tag{9.99} and then show that this can be written as V(R)≈−GMmR−GM2R3(I1+I2+I3−3IR),(9.100)V(\mathbf{R}) \approx -\frac{GMm}{R} - \frac{GM}{2R^3}(I_1 + I_2 + I_3 - 3I_R), \tag{9.100} where I1I_1, I2I_2, and I3I_3 are the moments around any three orthogonal axes (which we'll take to be principal axes in part (b)), and IRI_R is the moment around the axis along the R\mathbf{R} vector. ((b)) Consider now a planet with rotational symmetry around x^3\hat{x}_3, such as the earth which bulges at the equator due to the spinning. Using the result of Exercise 9.35, show that the potential in Eq. (9.100) can be written as V(R)≈−GMmR−GM2R3(I3−I)(1−3cos⁡2θ),(9.101)V(\mathbf{R}) \approx -\frac{GMm}{R} - \frac{GM}{2R^3}(I_3 - I)(1 - 3\cos^2\theta), \tag{9.101} where I≡I1=I2I \equiv I_1 = I_2, and θ\theta is the angle that R\mathbf{R} makes with x^3\hat{x}_3.

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