IB recommended length: 8 hours
Although we have already seem some of these concepts, in this section we will look deeper into the definitions and the relationships between work done, energy transferred and power - and how we can use these concepts to make predictions. We have looked at solving problems in mechanics using forces, but sometimes considering energetics (or a combination of both) is a better alternative.
Again, w e have met this concept before but it is worth restating. In classical physics energy is always conserved, i.e. in an isolated system (a system with no inputs of energy or matter from the outside) then the total amount of energy, in joules, is unchanged. More modern physics requires that we consider the overall mass-energy (calculated using E = mc² and similar equations) of a system, and that strictly this only applies in symmetrical situations (see Noether's theorem), but for the purposes of this section energy conversation is an inviolable assumption.
If a force moves an object, such as child pulling a sledge or bat hitting a ball, then we say that the force is doing work on the object. Specifically, we define work done, W, on a body by a constant force, F, as being equal to the product of the magnitude of the force and the displacement, s, applied in the direction of the line of displacement. The equation is:
W = F.s cos θ
where the work done is a scalar quantity, measured in joules, the force is measured in newtons, the displacement is measured in metres and cos θ indicates the deviation from the direction of the displacement. Theta, θ, can be in radians or degrees as it is the cosine of the angle that we need.
The work done is a transfer of energy, hence the units of joules (J). As we saw in the previous section, several forces can act on an object simultaneously, but it is the resultant, overall or net force on the object that predict what will happen to the object. The work done on a system by that resultant force is equal to the change in energy of the whole system.
Power is the rate of doing work, or how much work is done (or energy transferred) in a given time. The symbol for power is P and the unit of power is joules per second, J/s or watts, W. The main equation for power is therefore:
P = ∆W / ∆t
and, combing this equation with the equation for work done given above, we also have
P = Fv
where F is the applied force in newtons, N, and v is the velocity of the object in metres/second, m/s.
There are two overarching classifications of energy, potential and kinetic.
Kinetic energy was briefly introduced in the previous section when describing elastic and inelastic collisions, and is the energy an object has due to its movement. The equations for kinetic energy are:
Ek = 1/2.mv²
or in terms of momentum, p
Ek = p²/2m
where the kinetic energy, Ek, is in joules, mass, m, is in kilograms, linear velocity, v, is in m/s and momentum, p, is in kg.m/s.
Potential energy, Ep is the energy an object has due to its position in a field, (as we will see in topic D). We further classify the type of potential energy, and thus the equations we use to calculate it, depending on the situation and type of field.
Gravitational potential energy is the potential energy an object has due to its position in a gravitational field. For an object close to the surface of the Earth, then changing the objects height will change its gravitational potential energy by the equation:
∆Ep = mg∆h
where ∆Ep is the change in gravitational potential energy in joules, m is the mass of the object in kg, g is the local gravitational field strength of the Earth (= 9.81 N/kg) and ∆h is the change in the height of the object, measured perpendicular to the ground in metres.
Elastic potential energy, EH is the potential energy of an object that has been moved from an equilibrium position and is experiencing a restoring force. The standard example of this is a mass attached to a light stretched or compressed spring (light as in no mass, so we don't have to worry about the mass of the spring itself). The equation to calculate the elastic potential energy of such a system is:
EH = 1/2.k∆x²
where EH is the elastic potential energy of the object in joules, k is the spring constant, the force required to stretch the spring by 1m, in N/m, and x is the extension of the spring, the amount by which it is stretched, in m.
The sum of all of the kinetic energy, gravitational potential energy and elastic potential energy of a system of objects is called the mechanical energy of a system. In an isolated system, the total mechanical energy is conserved. Energy can be transferred between different forms but the total will remain the same. For example, a stationary ball might sit on top of a compressed spring. The spring has elastic potential energy. If the spring is released then the eleastic potential energy will be transferred into kinetic energy of the ball, which as it flies upward will be converted into gravitational potential energy. When all of the kinetic energy is transferred to gravitational potential energy the ball will be at the top of its trajectory, and will then convert from gravitational potential energy back to kinetic energy and then, assuming it lands back on the spring and there are no losses, back to elastic potential energy.
Of course in any real world situation there will be losses. Some energy might be converted to non-mechanical forms such as light or sound energy, but the main and final form of energy is thermal / degraded or internal energy. Internal energy is the energy inside an object, the kinetic energy of the moving particles and the potential energy due to the bonds between those particles, see the next topic for more details. Degraded energy is the dissipation of energy as heat into the surroundings and, eventually, the wider universe. Degraded energy is no longer available to do work.
We can describe, diagrammatically, the flow of energy and the energy changes in a system using a Sankey diagram. A basic Sankey diagram, or energy flow diagram, takes the form of a horizontal arrow, whose width is proportional to the amount of energy, or the power, in the system at a given point. Every time energy is transformed then the arrow splits as needed, usually to represent energy lost or degraded (in which case that split descends vertically). At all points the width of the arrows remain proportional to the energy flowing letting us easily trace both the path of the energy and what proportion is 'lost' or otherwise wasted at every stage.
The efficiency, η of a system is defined as the ratio of the 'useful' work done (or power) divided by the total work input (or power generated).
η = Eoutput / Einput = Poutput / Pinput
there are no units of efficiency, as it is a ratio, and clearly the definition of 'useful' is dependent on a given situation. Heat is normally a waste output for a lamp, but not if that lamp is designed to keep something warm.
A common example of a Sankey diagram in use, and efficiency calculations, involves the consumption of a fuel source. For example, the oil used to power a type of fossil-fuel power station or the petrol used in the internal combustion engine of a car.
Some fuel sources and processes are more efficient than others, which is an important consideration in their use, but another consideration is energy density. This is the amount of useful energy that can be stored in a given volume in joules per metre cubed, J/m³. Alternatively we may be concerned with the specific energy, which is the amount of energy per unit mass, normally joules per kg, J/kg. Uncompressed Hydrogen gas has a very high specific energy but a low energy density due to the low density of the gas. Gasoline has both a high energy density and a high specific energy, one reason why, despite its problems, it is a useful energy source.
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