7.4 Conservative Forces and Potential Energy

7.4 Conservative Forces and Potential Energy

• The equation is very similar to the kinematics equation, but it is more general and valid for any path regardless of whether the object moves with a constant acceleration or not.

• This is consistent with the observations made in Falling Objects that all objects fall at the same rate.
• Only the speed of the roller coaster is considered, there is no information about its direction at any point.
• This shows a general truth.
• The speed of a falling body depends on its initial speed and height, not on its mass or the path taken.
• The roller coaster will have the same final speed if it falls 20.0 m straight down or if it takes a more complicated path.
• The final speed in part (b) is more than in part (a) but less than 5.00 m/s.
• It is possible to find speed at any height by simply using the appropriate value at the point of interest.

• We have seen work that depends on the starting and ending points, and not on the path between, allowing us to define the idea of a simplified concept of potential energy.
• We can do the same thing for a few other forces, and we will see that this leads to a formal definition of the law of energy saving.

• This experiment can be used to study the conversion of potential energy into energy.
• Place a marble at the 10- cm position on the ruler and let it roll down the ruler to make an incline on a smooth, level surface.
• Measure the time it takes to roll one meter when it hits the level surface.
• Measure the times it takes to roll 1 m on the level surface by placing the marble at the 20- cm and 30- cm positions.
• The distance traveled by the marble is compared to the plot velocity.
• The plot shows that the marble's potential energy at the release point is proportional to the shape it is in.

• A marble is measured by rolling down a ruler.

• Some forces, such as weight, have special characteristics.
• When you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it.

• The spring has this characteristic because its force is conservative.
• A conservative force results in stored or potential energy.

• One example is the energy stored in a spring.
• We will see how conservativism is related to energy saving.

• A system has potential energy due to its shape or position.
• It is stored energy that can be recovered.

• A conservative force is one that depends on the beginning and end points of a motion and not on the path taken.

• Any conservative force can have a potential energy.
• The work done against a conservative force depends on the configuration and not the path followed.

• Let's get an expression for the potential energy stored in a spring.
• Hooke's law states that the work done to stretch or compress a spring must be calculated.
• The amount of deformation produced by a force will be replaced by the distance that the spring is stretched or compressed.
• Where is the spring's force constant, where is the force needed to stretch the spring?
• The force increases linearly from 0 to fully stretched position.

• The work done in stretching the spring is.
• The area under a graph is the work done by the force.

• The work done on the spring and the energy stored in it are represented by the potential energy.
• The potential energy of the spring is dependent on the stretch or squeeze in the final configuration.

• This work can be recovered because it is conservative and can be stored as potential energy in the spring.
• The work done or potential energy stored is.

• The equation has general validity even though it was derived from a special case.
• Potential energy can be stored in any elastic medium.
• Storage of energy is where the force constant of the particular system is.

• The guitar string has potential energy.
• As the string moves back and forth, the potential energy is converted back to potential.
• A very small fraction is dissipated as sound energy.

• When only conservative forces are involved, what form the work-energy theorem takes?
• This will lead us to the principle of energy conservativism.
• The net work done by all forces on a system is equal to the change in energy.

• The system loses potential energy if the conservative force works.

• The total energy is constant for any process involving conservative forces.

• This applies to the extent that all the forces are conservative.

• The track rises 0.180 m above the starting point.
• The force constant of the spring is 250.0 N/m.
• To find out how fast the car is going before it starts up the slope and how fast it is going at the top of the slope, you have to assume work done by friction to be negligible.

• A toy car is pushed up a slope by a spring.
• The potential energy in the spring is first converted to kinetic energy and then to a combination of the two as the car rises.

• The details of the path are unimportant because all forces are conservative.

• The spring force is conservative and can be used to conserve mechanical energy.

• When we start considering specific situations, this general statement becomes much simpler.
• First, we need to identify the initial and final conditions of the problem, then we need to solve it for an unknown.

• The problem is limited to conditions just before the car is released and after it leaves the spring.
• The initial height should be zero so that both are zero.

• In other words, the initial potential energy in the spring is converted to energy in the absence of friction.

• One way to find the speed at the top of the slope is to consider conditions just before the car is released and just after it reaches the top of the slope, completely ignoring everything else.