Work, Energy & Power
##### 2.5.1 Define Work

Work is not energy, it is a means of transferring energy by a force applied to a moving object. If the object does not move or the force is not in the direction of the motion then the force is not transferring energy to the object or we say “the force is not doing work on the object.” Mathematically we define work as:

(1)
\begin{align} W = \vec F \bullet s \end{align}

Or in IB world:

(2)
\begin{align} W = Fs \cos {\theta} \end{align}

Where W is the work, F is the force, s is the displacement and θ is the angle between the force and the displacement. The second formula is the one given by the IB and the one you are expected to understand. The first is the “real” equation it is a vector dot-product, but can be simplified to the second equation.

Note that if the force and displacement are perpendicular then no work is done by the force. If the angle is 180° then the force does negative work, an example of this would be a car traveling forward while the driver applies the brakes…

##### 2.5.2 Determine the work done by a non-constant force by interpreting a force displacement graph

The equation above work nicely if the force is constant, but in most cases the force is not constant… So we can either do some calculus or look at a graph (actually still doing calculus).

Imagine a force is applied to a cart in the same direction to the displacement. If the force varies with time and we plot the force vs. displacement:

In this case the formula can not be used, at least not simply. However, the area under the curve of the force vs. displacement is the work done on the cart by the force.

##### 2.5.4 Define kinetic energy

If a mass m is accelerated by a force F from initial velocity u and to a final velocity v. The displacement of the mass during the acceleration can be found by solving the following equation for s:

(3)
$$v^2 = u^2 +2as$$
(4)
\begin{align} s = {v^2 - u^2 \over 2a} \end{align}

Now the work done on the object is:

(5)
$$W = Fs$$
(6)
\begin{align} W = F \left ( \frac{v^2 - u^2} {2a} \right ) \end{align}

But the force is:

(7)
$$F = ma$$

If we assume that the object started with zero initial velocity, then we can write the work as:

W = ma \left ( {v^2 \over 2a} \right ) = \frac {1}{2}mv^2

This last expression is defined as the kinetic energy. Kinetic energy is the energy that an object has due solely to its velocity.

If we do not assume that the initial velocity is zero:

(8)
\begin{align} W = ma \left ( {v^2 - u^2 \over 2a} \right ) = \frac {1}{2}mv^2 - \frac {1}{2}mu^2 \end{align}

The first term on the right is the initial kinetic energy and the last term is the final kinetic energy. So the work done by accelerating the object is equal to the change in kinetic energy.

##### 2.5.5 Describe the concepts of gravitational potential energy and elastic potential energy

If an object of mass m is lifted vertically a distance h the work done on the object is:

(9)
$$W = Fs$$
(10)
$$W = Fh$$

The force needed to lift the object is:

(11)
$$F = mg$$

Or the weight of the object, we can then describe the work done as:

(12)
$$W = mgh$$

The last expression is the gravitational potential energy of the object. Potential energy is the energy an object has due solely to its position or configuration.

If an object is lifted from an initial height hI to a final height of hf. Then the displacement is:

(13)
$$s = h_f - h_i$$

And the work done on the object is:

(14)
$$W = mg(h_f - h_i)$$
(15)
$$W = mgh_f - mgh_i$$

Or in words the work done by lifting the object is equal to the change in the potential energy.

An object can also have elastic potential energy, if a spring is stretched or a rubber ball is deformed… In the case of the a spring the force of the spring is given by Hooke’s Law:

(16)
$$F = kx$$

So the work done is by displacing the spring a distance x is:

(17)
$$W = kx^2$$

This is also the potential energy due to the stretching the spring.

##### 2.5.6 State the principle of conservation of energy

Energy conservation is the principle that in a closed system energy is neither gained or lost. In an open system it is possible for energy to be added or lost. This definition can also be turned around. If a system is gaining or losing energy then it is an open system. The only truly open system in the universe, the earth is constantly gaining energy from the sun during the day and radiating the energy back out at night.

We can say that the total energy before an event is equal to the total energy after an event or:

(18)
$$Initial Energy = Final Energy$$

In terms of kinetic and potential energy:

(19)
$$KE_i + PE_i = KE_f + PE_f$$

So for a closed system, a falling ball is a good approximation, as the ball falls its potential energy decreases, but its kinetic energy increases at the same but opposite rate.

Note: Potential and kinetic energy are examples of mechanical energy, they are not the only types of energy.

This video shows two ways (or paths) that potential energy is converted into kinetic energy. The initial potential energies are both the same as both balls are dropped from the same height. The final kinetic energy is "shown" by the time the ball took to go through the photogate. The length of time taken to go through the photogate is only a function of speed and thus kinetic energy. Since the two times are the same the two final kinetic energies are the same.

##### 2.5.8 Define power

Power is the rate that work is done or the rate that energy is transferred.

(20)
\begin{align} P = {W \over t} \end{align}

The unit of power is Watts, $Watt = {Nm \over s}$. This is the same unit as on your light bulbs and electrical appliances.

We can also rewrite the power as:

(21)
\begin{align} P = {Fs \over t} = Fv \end{align}

Where F is the force applied, s is the displacement, t is the time and v is the velocity.

##### 2.5.9 Define and apply the concept of efficiency

When work is done on an object, sometimes the energy is converted into unwanted or non-useful forms (often heat). The ratio of useful energy to the amount of energy applied is the efficiency, it can also be defined in terms of power:

(22)
\begin{align} Efficiency \% = {Useful Energy \over Total Energy} \times 100\% = {Useful Power \over Total Power} \times 100 \% \end{align}