Conservative forces When the forces can be derived from a potential energy function, v, we say e forces are con such cases, we have that F=-Vv, and the work and energy relation in equation 2 takes a particularly simple form. Recall that a necessary, but not sufficient, condition for a force to be conservative is that it must be a function of position only, i. e. F(r)and V(r). Common examples of conservative forces are gravity (a constant force independent of the height), gravitational attraction between two bodies(a force inversely proportional to the squared distance between the bodies), and the force of a perfectly elastic spn: sely The work done by a conservative force between position ri and r2 is Thus, if we call Wis the work done by all the external forces which are non conservative, we can write the Ti+Vi+Wi=T2 Of course, if all the forces that do work are conservative, we obtain conservation of total energy, which can Gravity Potential for a Rigid Body In this case, the potential Vi associated with particle i is simply Vi=migzi, where zi is the height of particle i above some reference height. The force acting on particle i will then be Fi=-VVi. The work done on the whole body will be f∫ V)1-(V)2)=∑mg(2)-(2 There the gravity potential for the rigid body is simply, mi92i=920 where zg is the z coordinate of the center of mass. Example Cylinder on a Ramp We consider a homogeneous cylinder released from rest at the top of a ramp of angle o, and use conservation of energy to derive an expression for the velocity of the cylinderConservative Forces When the forces can be derived from a potential energy function, V , we say the forces are conservative. In such cases, we have that F = −∇V , and the work and energy relation in equation 2 takes a particularly simple form. Recall that a necessary, but not sufficient, condition for a force to be conservative is that it must be a function of position only, i.e. F(r) and V (r). Common examples of conservative forces are gravity (a constant force independent of the height), gravitational attraction between two bodies (a force inversely proportional to the squared distance between the bodies), and the force of a perfectly elastic spring. The work done by a conservative force between position r1 and r2 is W1−2 = Z r2 r1 F · dr = [−V ] r2 r1 = V (r1) − V (r2) = V1 − V2 . Thus, if we call WNC 1−2 the work done by all the external forces which are non conservative, we can write the general expression, T1 + V1 + WNC 1−2 = T2 + V2 . Of course, if all the forces that do work are conservative, we obtain conservation of total energy, which can be expressed as, T + V = constant . Gravity Potential for a Rigid Body In this case, the potential Vi associated with particle i is simply Vi = migzi , where zi is the height of particle i above some reference height. The force acting on particle i will then be Fi = −∇Vi . The work done on the whole body will be Xn i=1 Z r 2 i r 1 i fi · dri = Xn i=1 ((Vi)1 − (Vi)2) = Xn i=1 mig((zi)1 − (zi)2 = V1 − V2 , where the gravity potential for the rigid body is simply, V = Xn i=1 migzi = mgzG , where zG is the z coordinate of the center of mass. Example Cylinder on a Ramp We consider a homogeneous cylinder released from rest at the top of a ramp of angle φ, and use conservation of energy to derive an expression for the velocity of the cylinder. 4