Chapter 15459P15.56The kinetic energy of the ball is KmvI=+121222Ω,where Ωis the rotation rate of the ball about itscenter of mass. Since the center of the ball movesalong a circle of radius 4R, its displacement fromequilibrium is sR=4afθand its speed isvdsdtRddt==FHGIKJ4. Also, since the ball rolls withoutslipping,vdsdtRΩsoΩ==FHGIKJvRddt4The kinetic energy is thenRddtmRddtmRddt=FHGIKJ+FHGIKJFHGIKJ=FHGIKJ124122541121022222θθh5R R sFIG. P15.56When the ball has an angular displacement , its center is distance hR=−41coshigher thanwhen at the equilibrium position. Thus, the potential energy is UmghmgRg−41cos. For smallangles, 122−≈cos(see Appendix B). Hence, gRg≈22, and the total energy isEKU
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