8 Chapter 8: Conservative and Non-Conservative Forces
8 Chapter 8: Conservative and Non-Conservative Forces
Textbook Section 8.2: Conservative and Non-Conservative Forces
Section 8.1: Conservative Forces
The amount of work done on an object only depends on where it started and where it ended; it does not depend on the path you took from the initial to final place. Bouncing a book up and down before putting it back on the shelf might make you think you’re expending energy, but according to physics, no net work has been done on the book if it ended up where it started. Forces that work this way (only depend on initial and final points, not the path between) are called conservative forces. Gravity, other forms of potential energy (such as spring forces) are conservative forces. Friction is an example of a non-conservative force; because friction depends on how much contact you have had with the ground, the longer the path means the more energy is lost to fiction.
Section 8.2: Spring Potential Energy
As you stretch a string away from its equilibrium position (the position it returns to when not stretched or compressed) you must continually add energy to the system. At first, the spring stretches easily, but the longer it gets, the larger the required force to stretch it further. While still a conservative system (you put energy into the spring to compress it, which is then transferred to another object when released), it’s not a constant force relationship. For this reason, the area under the curve is a triangle, not a rectangle, and the work done on the spring is given by
The force needed to stretch a spring is given by , where is the spring constant (a constant value for each spring, small if the spring is easy to stretch and large if the spring is stiff, given in units of ) and or is the distance the spring is compressed or stretched away from its equilibrium position (a of zero means the spring is in equilibrium).
The potential energy stored in a spring by extending or compressing it some distance is given by
The force required to stretch a spring some distance is given by
We now have yet another conservative energy term to add to our conservation of energy equation. Because all the energy we have initially must be all the energy we have at the end (simply transformed into different forms), we can add spring potential energy to the equation to get
where is the initial extension of the spring and is the final extension of the spring. Note that when using the above equation, we are talking about two distinct moments in time, initial and final. Everything on the left happens at the initial moment, everything on the right happens at the final moment. If a type of energy is not changing, you can cancel it from both sides (for example, object is not moving at beginning or end — cancel ; no springs are involved — cancel ; object does not change height — cancel ).
Section 8.3: Calculating Non-Conservative Forces
Textbook Section 8.3: Conservation
Non-conservative forces are forces that rely on the path you