For the motion of a projectile, the total energy is then 1 mu+ mgz= constant Since v and v, remain constant, we also have -muz +mgz= constant gravity In a central gravity field and so, taking I(r→∞)=0, Mn Spring Fo orce The force supported by a spring is F==kr. The elastic potential energy of the spring is the work done on it to deform it an amount z. Thus, we have -kr dxr If the deformation, either tensile or compressive, increases from 1 to z2 during the motion, then the change in potential energy of the spring is the difference between its final and initial values, or △V=k(x2-x2) Exercise Non-conservative forces Consider a block sliding on a horizontal surface. Is the friction force conservative? Explain why? Give examples of non-conservative forces Note Path independent forces for which there is no potential(Optional) The condition F=-VV is sufficient, bu ut not necessa ary for the work to be path independent. An important example is the magnetostatic force F=qU x B, where the work F.dr= F. udt is zero(the force is alway perpendicular to u). Thus, for a general velocity-dependent electro-magnetic force, F=q(E+uxB),where only B is irrotational, the work is still path independent, and the total energy(mu2/2+qo)is conserveFor the motion of a projectile, the total energy is then E = 1 2 mv2 + mgz = constant . Since vx and vy remain constant, we also have 1 2 mv2 z + mgz = constant. Gravity In a central gravity field F = −G Mm r 2 er = −∇(−G Mm r ) , and so, taking V (r → ∞) = 0, V = −G Mm r . Spring Force The force supported by a spring is F = −kx. The elastic potential energy of the spring is the work done on it to deform it an amount x. Thus, we have V = − Z x 0 −kx dx = 1 2 kx2 . If the deformation, either tensile or compressive, increases from x1 to x2 during the motion, then the change in potential energy of the spring is the difference between its final and initial values, or, ∆V = 1 2 k(x 2 2 − x 2 1 ) . Exercise Non-conservative Forces Consider a block sliding on a horizontal surface. Is the friction force conservative? Explain why? Give examples of non-conservative forces. Note Path independent forces for which there is no potential (Optional) The condition F = −∇V is sufficient, but not necessary for the work to be path independent. An important example is the magnetostatic force F = qv × B, where the work F · dr = F · vdt is zero (the force is always perpendicular to v). Thus, for a general velocity-dependent electro-magnetic force, F = q(E +v×B), where only B is irrotational, the work is still path independent, and the total energy (mv2/2 + qφ) is conserved. 4