CIE A Level Maths: Mechanics復習筆記4.1.3 Energy Principles
Conservation of Energy
What is conservation of energy?
- The?principle of conservation of mechanical energy?states that when a particle is moving and?no external forces?other than gravity act to do?work?on it, then the total sum of its?kinetic energy?and the?gravitational potential energy?will remain constant
- Depending on how the particle is moving, either kinetic energy will be transferred to gravitational potential energy or gravitational potential energy will be transferred to kinetic energy
- If a particle’s height increases it will?gain GPE?and so must?lose KE?(and must be slowing down)
- Increase in GPE = Decrease in KE
- If a particle’s speed increases, it will?gain KE?and so must?lose GPE?(and must be moving downwards)
- Increase in KE = Decrease in GPE
How is conservation of energy used on an inclined plane?
- If the particle is travelling on an?inclined plane?it is important to make sure you find the?vertical distance?before calculating the change in?GPE
It is important to make sure that the?only force?acting on the particle is gravity before applying the?principle of conservation of mechanical energy
- Check that there is no friction by making sure that the plane is a?smooth?plane
- The?normal reaction force,?R, will be acting perpendicular to the plane and so will do?no work?on the particle
- Conservation of energy is often used on a particle that has been?projected?up a smooth slope with no other external forces acting on it, as it moves up the slope it?gains GPE?but?loses KE
How is conservation of energy used with projectiles?
- Sometimes problems involving?projectiles?can be solved using energy principles
- There are?no external forces?acting on the particle other than?gravity
- The total energy will not change whilst the particle is in motion
- Energy principles are useful to use instead of 'suvat'?if mass is known or needed
Worked Example
Exam Tip
- Look out for terms such as?held at rest?or?comes to rest, meaning that the start or finish speed is zero. Check that units are in metres, seconds and kilograms before carrying out any question.
Work-Energy Principle
What is the work - energy principle?
- If one or more?external forces?other than gravity act on a particle during its motion then its total energy can?change
- The?work - energy principle?states that the?'the net work done by external forces = the total change in kinetic energy'
- In problems with more than one force acting on the particle, the?net work done?by all external forces, including gravity?will be equal to the?change in kinetic energy?only
- Rearranging this such that the work done by gravity is on the other side of the equation gives the result 'work done?by all forces?except weight?is equal to the?change in?mechanical energy'
- If a particle has to overcome a?resistive force?such as?friction?it will?lose kinetic energy
- The?work - energy principle?is particularly useful in problems when we do not know that motion is in a straight line
- As long as we know the?change in vertical height?between the start and finish the work – energy principle can be applied
- The particle does not need to have taken a direct route to get from the start to the finish point
- The work - energy principle can also be used in problems where?acceleration?is?not constant
- This makes it particularly useful as it can sometimes be used when?is not possible
- If it is not known that acceleration is constant then you should use energy methods to solve the problem rather than the 'suvat' equations
How is the work – energy principle used on an inclined plane?
- If the particle is travelling on an?inclined plane?it is important to make sure you find the?vertical distance?before calculating the change in?GPE
- If the surface is defined as a?rough?surface then the particle will have to overcome friction and so will have to do?work
- This is a hint that it is a problem involving the work – energy principle
Worked Example
A dog uses a constant force of 5 N to push its toy of mass 500 grams up a rough slope inclined at 20° ?to the horizontal. The toy starts from rest and after moving 10 metres up the line of greatest slope it is travelling at 1 m s-1. ?Given that the force is acting parallel to the slope, find the coefficient of friction between the toy and the slope.
Exam Tip
- Look out for questions that don’t explicitly say that motion is linear or acceleration is constant, these are hints that you should use the principle of conservation of mechanical energy instead of the 'suvat'?equations.?If you’re unsure try to use the energy methods first to solve a problem.
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