Conservation of Energy

Conservation of Energy

There are two categories of forces we will consider, conservative and non-conservative.

A conservative force results in stored or potential energy and we can define a potential energy (\(E_p\)) for any conservative force.

Gravity is a conservative force and we studied gravitational potential energy earlier. We now have all the concepts we need to actually deduce this ourselves. Let us consider pushing a ball up a number of different slopes.

Three different slopes are shown, all rising to a height of \(h\). The imaginary right-angled triangle is shown for each slope. \(d\) is the length of the slope. \(\alpha\) is the angle the slope makes with the horizontal.

The slope, of length \(d\) is the hypotenuse of an imaginary right-angled triangle. The work done by gravity while pushing a ball of mass, \(m\), up each of the slopes can be calculated. We know that the component of the gravitational force parallel to the slope is \(\vec{F}_{gx}=\vec{F}_g\sin\alpha\) down the slope.

The work done by gravity when the force \(\vec{F}\) pushes the ball up the slope will benegative because the direction of the motion and \(\vec{F}_g\sin\alpha\) are opposite.

This final result is independent of the angle of the slope. This is because\(\sin\alpha=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{h}{d}\) and so the distance cancels out. If the ball moves down the slope the only change is the sign, the work done by gravitystill only depends on the change in height. This is why mechanical energy includes gravitational potential energy and is conserved. If an object goes upa distance \(h\) gravity does negative work, if it moves back down\(h\) gravity does positive work, but the absolute amount of work is the sameso you `get it back', no matter what path you take!

This means that the work done by gravity will be same for the ball moving up any of the slopes because the end positionis at the same height. The different slopes do not end in exactly the same position in the picture. If we break each slope into two sectionsas show in the figure below then we have 3 different paths to precisely the same end-point. In this case the total work done by gravity along each path is the sum of the workdone on each piece which is just related to the height. The total work done is related to the total height.

Three different paths that lead from the same start-point to the same-end point. Each path leads to the same overall change in height, \(h\), and, therefore, the same work done by gravity.

There are other examples, 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. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring.

The total work done by a conservative force results in a change in potential energy, \(\Delta E_p\). If the conservative force does positive work then the change in potential energy is negative. Therefore:

Friction is a good example of a non-conservative force because if removes energy from the system so the amount of mechanical energy is not conserved. Non-conservative forces can also do positive work thereby increasing the total mechanical energy of the system.

The energy transferred to overcome friction depends on the distance covered and is converted to thermal energy which can't be recovered by the system.