(downward
direction) by applying an opposite force to move an "m" mass object
upward, for an "h" distance, we increase the gravitational potential
energy of the object. The amount of gained potential energy as usual is equal to
the work done:In many processes there is no Kinetic energy gain when work is done. Think for instance of the action of compressing a spring or of the action of lifting a book from the floor to a higher position on a table.
All these actions need work to be done but both the spring and the book were at rest before , and stay at rest after the work is done on them: there’s no change in their Kinetic energy.
In these examples the effect of the work is to increase not the Kinetic, but the POTENTIAL ENERGY of the system.
Potential Energy can be defined every time we change the position of a system that is being acted on by a specific force. Potential Energy can be defined every time we do work to "move" a system from a certain state to a different state in which the system can "potentially" develop all the stored energy due to the work done, in order to come back to its original state. For this reason, it is possible to define Potential Energy if, and only if, we do work against forces which are different from frictional forces. As a matter of fact, frictional forces involve a loss of the work done on a system, by energy dissipation (heat) in the neighbouring environment. In other words: It’s possible to define Potential Energy only in the case of conservative forces. The Mathematical expression for the Potential Energy depends on the particular conservative force we work against.
If we work against the gravitational force(downward
direction) by applying an opposite force to move an "m" mass object
upward, for an "h" distance, we increase the gravitational potential
energy of the object. The amount of gained potential energy as usual is equal to
the work done:
Gravitational potential
Energy
(see the figure and the sample exercise on page 101 in the text book)
If we compress or stretch a spring, we do work against an opposing elastic force.
Robert Hooke (1635-1703) discovered that the force exerted by a spring is proportional to the distance the spring is stretched or compressed.
Hooke’s Law
is the proportionality
constant and describes the spring stiffness. Its value depends on the particular
spring we use: A stiff spring has a large spring constant and vice-versa.
is the distance from the
spring equilibrium position.
In the Hooke law we have a minus sign because the elastic force exerted by the spring is always in the opposite direction of the stress. For example if the spring is stretched to the right, the spring reacts by pushing back to the left.
To increase the potential energy of the spring we have to work against the spring, this means we have to apply a force against the spring in order to compress or to stretch it for a distance "x".
so the potential energy will be:
Elastic potential Energy
Note that, the average force exerted can be computed by making the sum between the final and the initial force just because the force has a linear dependence on distance. In general this is not true for all kinds of forces.