1. Physics 141Mechanics Lecture 9 Conservation of EnergyYongli Gao Consider a particle moving in 1-D under a force F(x). As it moves from x i to x f the work done by the force is If we define potential energy as which leads to or, the sum of potential energy and kinetic energy is a constant of the motion. This is one of the most fundamental principle of physics: conservation of energy. Also,

2. Gravitational Potential Energy If we choose the vertical to be the y-axis, the gravitational force is W=-mgj, and the gravitational potential energy is At reference point y =0, the potential energy is zero. In fact, we can choose any one point as the reference point, and stick to it afterward. If we drop a particle at rest from y i =h to y f =0, from conservation of energy, we have

3. Demonstration: Loop the Loop A slope is connected at the bottom to a vertical loop of radius R. Where should we put a particle on the slope so that it’ll remain in contact to the top of the loop when sliding down? From energy conservation At the top of the loop, the criterion is N=0, and the gravity alone provides the needed centripetal force mgmg N

4. Example: Workdone by a Variable Force A particle of mass m is attached to a string of l. If one pushes m by a horizontal force just barely enough to move it, what would be the work done on m when l makes an angle  f to the vertical? Solution: Since it’s barely moving, we can take it that it is almost in equilibrium,  T F mgmg

5. Discussion: We see that the work done by the applied horizontal force is in fact the increase of the gravitational potential energy. It is obvious from energy conservation since there’s no friction, the tension T does no work since it is always perpendicular to the displacement dr, and the kinetic energy hasn’t changed. We may also solved the problem by realizing

6. Demonstration: Conservation of Energy We have a pendulum of length l and mass m, and an adjustable peg in the path of the string. If we place the peg at d below the pivot, what should be the angle  we must lift the pendulum to such that the mass m will pass the highest point defined by the peg? At the top, the speed must be such that gravity only supply the centripetal force Energy conservation

7. Position Dependent Force: a Spring The most common example of position dependent force is a spring. In fact, for any motion in the vicinity of equilibrium, the motion can be approximated as that of a particle attached to a spring. The spring force is described by Hooke’s law F=-kx where x is the displacement of the particle from the unstretched position (or equilibrium position), k is a positive constant called spring constant. It’s unit in SI is N/m. Note the negative sign. It means that the force is always in the direction opposite to the displacement to bring the particle back to the equilibrium position. For this reason, a spring force is oftem called a restoring force.

8. Demonstration: Spring and Hooke’s Law We’ll see how the spring force is described by Hooke’s law F=-kx

9. Elastic Potential Energy What is the work done by a spring if the displacement of the particle attached to the spring changes from x i to x f ? If we choose the reference point at U(x=0)=0, the elastic potential energy is

10. Example: Bungee-Cord Jumper If the starting point of bungee-cord jump is h from the ground, and the cord is of relaxed length L<h and elastic coefficient k, what is the maximum mass m of the jumper the apparatus can handle? Solution: Conservation of energy K i =0 and to be safe K f =0. We get the maximum safe mass In the real world, you may give yourself a leeway by putting in some safety factor to count for a possible initial jump or the fact the person is not a point.

11. Conservative Forces Only a conservative force can be associated with a potential. The most important feature of a conservative force is that the work done by it depends only on the initial and final positions, not the path it takes. Or, along any closed loop, the work done is zero. Not all forces are conservative. The work done by a non-conservative force does depend on the path. One example is friction. When dragging a box on the floor, the work done by friction depends on the path. Here the mechanic energy is turned to heat. AB 1 2 3

12. Example: Non-Conservative Force Suppose F=-yi+xj, calculate the work done by the force from (0,0) to (2,3) by taking a) (0,0) to (0,3) to (2,3); b) (0,0) to (2,0) to (2,3). Solution: a) Fdr=(-yi+xj)(dxi+dyj)=-ydx+xdy b) x y (2,3) (2,0) (0,3)