238 Potential EnergyP8.54The gain in internal energy due to friction represents a loss in mechanical energy that must be equalto the change in the kinetic energy plus the change in the potential energy.Therefore,−=+−µθθkmgxKkxmgxcossin∆122and since vvif==0, ∆K=0.Thus,−°=−°µk2 00 9 8037 00 200100 0 20022 00 9 8037 00 2002..cos.sinafafafafafafand we find k=0115.. Note that in the above we had a gainin elastic potential energy for thespring and a lossin gravitational potential energy.P8.55(a)Since no nonconservative work is done, ∆E=0Also ∆K=0therefore, UU=where Umgxi=sinbgand Ukxf=1222.00 kgk=100 N/mFIG. P8.55Substituting values yields
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