20 1. Introduction 1.6. Exercises 1.6.1. Descent time down a cycloidal curve. Show that the de- scent time down the cycloidal curve x(φ) = a + R(φ − sin φ) , y(φ) = ya − R(1 − cos φ) (1.48) is T = R g φb , (1.49) where φb is the angle φ corresponding to the point B = (b, yb). What is the descent time to the lowest point on the cycloid? 1.6.2. Complementary curves of descent. The authors Mungan and Lipscombe (2013) recently introduced the term complementary curves of descent to describe curves that have identical descent times. (a) Determine the descent time for a straight line (shown in bold in Figure 1.11). (b) Rewrite integral (1.5) in polar coordinates assuming, for conve- nience, that θ increases clockwise. (c) Determine the descent time for the lower portion of the leminis- cate r = 2 c √ sin θ cos θ (1.50) (shown in bold in Figure 1.11). Hint: d dθ cos1/4 θ sin1/4 θ = − 1 4 cos−3/4 θ sin−5/4 θ . (1.51) (d) Verify that the lemniscate is complementary to the straight line. 1.6.3. Potential energy due to a spherical shell. The gravita- tional potential energy between two point masses, M and m, sepa- rated by a distance r is V (r) = − GMm r , (1.52) where G is the universal gravitational constant. Calculate the potential energy of mass m at point P due to the gravitational attraction of a thin homogeneous spherical shell of mass
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