Energy, conservation and efficiency (Topic 2D)
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Kinetic energy is the energy possessed by an object due to its motion. Kinetic energy is a scalar quantity that has a magnitude but no direction. The formula that describes the kinetic energy is:
Where:
- is the mass in kilograms (), and
- is the speed in metres per second ().
The kinetic energy of an object is measured in joules () and is directly proportional to its mass and to the square of its speed.
For example, doubling the mass of an object moving at the same speed would double its kinetic energy. Alternatively, doubling the velocity of the same object would quadruple its kinetic energy.
Gravitational potential energy (GPE) is the energy possessed by an object due to its position in a gravitational field. GPE is a scalar quantity that has a magnitude but no direction:
Where:
- is the mass in kilograms (),
- is the height above an arbitrary reference level in metres (), and
- is the gravitational field strength in metres per second squared ().
The gravitational potential energy of an object is measured in joules (). It can be positive, negative, or zero.
In the diagram above a ball at:
- the zero reference has no gravitational potential energy,
- a ball at a distance above the reference has a gravitational potential energy equal to , and
- a ball at a distance below the reference has a gravitational potential energy equal to .
Since gravitational potential energy is defined from a reference point, it is often required to determine the change in the gravitational potential energy between two positions.
The diagram below shows a ball thrown upward. The ball starts from A, rises to its maximum height at B, then falls back towards its initial vertical position at C. Taking the zero reference at the level of the hand:
- The gravitational potential energy at A and C is equal to zero
- The gravitational potential energy at B is equal to
The change in gravitational potential energy from point A to point B:
Now the change in gravitational potential energy from point B to point C:
This implies that any rising object gains gravitational potential energy while any falling object loses gravitational potential energy.
Question walkthrough
GPE Change Independent of Zero Reference
Calculates gravitational potential energy at three points using two different zero references, showing that the change in GPE between any two points is the same regardless of the reference chosen.
The exchange between kinetic energy and gravitational potential energy occurs for an object moving in a gravitational field. Examples include:
- An object that falls from the roof of a building loses gravitational potential energy and gains kinetic energy.
- A swing at its highest position, moving towards its lowest position, loses gravitational potential energy while gaining kinetic energy.
- When a ball is thrown up with an initial velocity, it loses kinetic energy as it ascends and gains gravitational potential energy.
For an object moving in a gravitational field, in the absence of frictional forces, the gain in one form of energy is equal to the loss in the other:
If a ball at A possesses of gravitational potential energy and falls, it loses of by the time it reaches B. This loss in is converted to kinetic energy. Therefore, the ball possesses of when it reaches the ground.
Although systems in a gravitational field may look different, the way that their energy changes between kinetic and gravitational in the absence of frictional forces is the same.
General statements can be made about objects in a gravitational field released from a raised level or projected upwards from a lower level:
- At maximum height, an object’s kinetic energy is minimal, and its gravitational potential energy is at its maximum.
- At ground level, an object’s gravitational potential energy is minimal, and its kinetic energy is at its maximum.
- At intermediate heights, kinetic energy and gravitational potential energy will vary, but their sum remains constant, which can be written as:
This is a statement of the law of energy conservation.
Question walkthrough
Energy Conservation on a Frictionless Slope
Uses GPE = mgh and KE = ½mv² to track energy conversion for a frictionless skier's descent, then finds the resulting speed at the bottom of the slope.
The principle of conservation of energy states that energy cannot be created or destroyed; it can only be transferred or transformed from one form to another.
For example, for an ideal pendulum, gravitational potential energy is converted to kinetic energy and vice versa during each swing, but the total energy stays the same.
The principle of conservation of energy is best illustrated in an isolated system.
- Isolated system: energy or matter cannot be exchanged with the surroundings. In an isolated system, the total energy remains constant.
- Closed system: matter cannot be exchanged with the surroundings, but energy can be.
- Open system: energy and matter can be exchanged with the surroundings.
Wasted energy is simply defined as energy that is not useful. Wasted forms of energy typically include heat, light and sound.
For instance, in any real mechanical system, useful energy output is always less than the total energy input because friction between a machine’s moving parts generates heat and sound.
Question walkthrough
Finding Work Done Against Friction
Uses the work-energy principle to find the energy lost to friction when a force accelerates an object over a known distance, then finds the corresponding average frictional force.
Energy exists in many forms. Common examples of energy pathways include:
Energy can be transferred from one object to another or from one form to another, in accordance with the principle of conservation of energy.
Energy exists in many forms. Common examples of energy stores include:
Energy can be transferred from one object to another or from one form to another, according to the principle of conservation of energy.
Question walkthrough
Energy Transfers in a Mass-Spring System
Describes how elastic potential energy converts into kinetic and gravitational potential energy as a vertically oscillating mass moves from its stretched position back toward equilibrium.
Doing work on an object transfers energy to it. For example, lifting a weight increases its gravitational potential energy, which equals the work done in lifting it.
The greater the magnitude of a force or the distance over which it is applied, the more energy is transferred:
Question walkthrough
Finding Height Gained from Energy Transfer
Uses the work done by a piston force to split transferred energy between heating water and lifting a platform and scooter, then finds the height gained using W = mgh.
Power is defined as the rate of doing work. In other words, power is the amount of work done per unit of time:
Where:
- is power in watts ,
- is work in joules , and
- is time in seconds .
Another common unit of power is the joules per second
Differentiating between energy, work, and power can be challenging. The table below compares the three quantities and provides an example of each.
Question walkthrough
Calculating a Lift's Work and Power
Uses a lift's constant driving force and floor height to find distance travelled, then applies W = Fd and P = W/t to calculate work done and power output.
Efficiency is a term used to describe how well energy is converted in a mechanical system:
- A system with high efficiency converts most of the input energy into useful output energy.
- A system with low efficiency converts most of the input energy into wasted output energy.
Efficiency can be expressed as a decimal between 0 and 1 or as a percentage between 0% and 100%. A system with an efficiency of zero wastes all the input energy, and an efficiency of one (100%) is considered ideal. It converts all the input energy into useful energy. In reality, a system can never be 100% efficient.
Mathematically, the efficiency of a mechanical system is defined as the ratio of the useful output energy to the input energy:
To get the efficiency in percentage form, we apply the ratio below:
It is important to note that effiency can be anything between zero and one or 0% and 100%.
The efficiency can also be described in terms of power. Efficiency can also be defined as the ratio of the output power to the input power. In decimal form, the efficiency is:
And in percentage form, the efficiency is:
Question walkthrough
Calculating Elevator Motor Power and Efficiency
Calculates an elevator motor's output and input energy, its input and output power, and verifies the result matches the stated efficiency.