Physics 201 Potential Energy Conservative and Nonconservative Forces Conservation of Energy Changes of Energy in presence of both Conservative and Nonconservative Forces Energy Diagrams 7: Potential Energy and Conservation of Energy

Potential Energy Work done by the spring force on an object only depends on the coordinates of the “end” points of the displacement of the spring and of the object (while it is in contact with the spring) x=0

Forces that do work with this property are called CONSERVATIVE FORCES and do not depend on the path that the object that experiences the force follows  do not depend on length of path between initial and final point NONCONSERVATIVE FORCES do not have this property  do depend on length of path between initial and final point

x=0 F Work done by spring on block while spring is compressed from x i  0 to x f  m (for example) W  1 2 k  2 J  0 x=0.25m

x=0 If block and spring are let go then block will move to right from x i  m to x f  0 m work done as spring expands back to its equilibrium length by spring on block W  1 2 k0.25  2 J  0 by the Work -Energy theorem W(by spring)=  K(of block)= 1 2 m v f 2  1 2 m v i 2

x=0 v The initial velocity is v i =0 m s The final velocity is v f  k m 0.25  If no further forces act on the block (of mass m) then it will continue to move at this velocity m s

We see that work done by the spring on the block when it expands back to its equilibrium position is transformed into kinetic energy (motional energy) of the block. The amount of energy transformed to the kinetic energy of the block was exactly the negative of the work done on the block by the spring while the spring was compressed and then kept fixed. Once the spring+block was released the negative of this work became the kinetic energy of the block We thus say that this work was stored as POTENTIAL ENERGY (due to the spring) in the block and define the change of POTENTIAL ENERGY of the block (due to the spring) as the negative of the work done by the spring on the block

 U  U f  U i  W change of potential energy of the block =  work done by the spring force on the block  U f  U i  1 2 kx f 2  1 2 i 2      1 2 f 2  1 2 i 2  U i  1 2 i 2 ; U f  1 2 f 2 Thus we say that the potential energy (due to the spring force) of the block at displacement x is Ux  = 1 2 kx 2 The definition of potential energy only makes sense for conservative forces

notice that Fx   kx  dUx  dx  d 1 2 kx 2     dx This is true in general for all conservative forces Fx   dUx  dx Energy Diagrams U x x1x1 x2x2 x3x3 U(x)

x 1 =stable equilibrium position x 2 =unstable equilibrium position x 3 =stable equilibrium position for spring U x

Gravitational Potential Energy work done by gravity on object of mass m when object is lifted through height h and we approximate the force (near the surface of the earth) of gravity by a constant weight force W W  W  h  mgh Thus change of(gravitational ) potential energy of the  U  W  mgh Gravity is a conservative force, thus the path through which the mass is lifted does not effect the work done by gravity on the mass. mass is

Conservation of Energy The total Mechanical Energy of an object is defined to be E tot  U  K U is due to all conservative forces acting on object If we only consider conservative forces effecting the object W  K  U   K  U  0  K  U   0  E tot  0 Law of conservation of total mechanical energy

If we consider both conservative and non conservative forces W tot  W con.  W noncon.  K W con  U tot  W noncon  K  U tot  E W noncon.  E tot

Generalization of conservation of mechanical energy We can define U =  W noncon  int where U int is called the internal energy of an object due to its internal construction i.e. atoms and molecules and the forces between them.

Then  K  W tot  W con.  W noncon.  U con  U int.  K  U con  U int.  0  K  U con  U int.   0  E tot.mech  U int.   0  E tot  0

Conservative Forces Gravity Spring Electric Nonconservative Forces Friction frictional force cause change of internal energy that are manifested as heating of the object