WORK Work provides a means of determining the motion of an object when the force applied to it is known as a function of position. For example, the force exerted by a spring varies with position: F=-kx where k is the spring constant and x is the displacement from equilibrium.
Recall Our Discussion of the Concept of Work Work has no direction associated with it (it is a scalar). However, work can still be positive or negative. Work done by a force is positive if the force has a component (or is totally) in the direction of the displacement.
CONSERVATIVE FORCES A force is conservative if the work it does on a particle that moves through a closed path is zero. Otherwise, the force is nonconservative. Conservative forces include: gravitational force and restoring force of spring. Nonconservative forces include: friction, pushes and pulls by a person. FgFg d
CONSERVATIVE FORCES If a force is conservative, then the work it does on a particle that moves between two points is the same for all paths connecting those points. This is handy to know because it means that we can indirectly calculate the work done along a complicated path by calculating the work done along a simple (for example, linear) path.
Work Done by Conservative Forces is of Special Interest The work “done” in the course of a motion, is “undone” in if you move back. This encourages us to define another kind of energy (as opposed to kinetic energy) - a “stored” energy associated with conservative forces. We call this new type of energy potential energy and define it as follows: U = – W c FgFg d
Potential Energy Associated with the Gravitational Force
Potential Energy Associated with the Spring Force We know (or should know) from our homework,
Tying Together What We Know about Work and Energy U = – W c W net = K So, under the condition that there are only conservative forces present : W net = W c In that case, K = – U K + U = 0
The “Bottom Line” E i = E f K i + U i = K f + U f The “Total Mechanical Energy” of a System is the sum of Kinetic and Potential energies. This is what is “conserved” or constant. u Gravitational force: U= mgh u Restoring force of a spring: U =1/2kx 2 u (KE=1/2mv 2 )
An Example A 70 kg skate boarder is moving at 8 m/s on flat stretch of road. If the skate boarder now encounters a hill which makes an angle of 10 o with the horizontal, how much further up the road will the he be able to go without additional pushing? Ignore Friction.
10 o h d KE i +U i =KE f +U f (only conservative forces) so KE i + 0 = 0+U f (U i =0 and KE f =0) 1/2mv 2 = mgh 1/2v 2 = gh h = v 2 /(2g) = 8 2 /(2*9.8) = 3.26 m h/d = Sin 10 o d = 18.8 m