S3.7 Energy conservation
[A] Conservation laws in physics
If the total amount of some quantity, Q say, does not change with time, we say it is conserved. Mathematically, we can express this as
- Conservation laws are fundamental laws of nature.
- Conservation laws are useful because:
- knowing a quantity is conserved is a very valuable aid to problem solving.
- they provide restrictions on the form of hypotheses that can be advanced.
[B] Energy Conservation
Energy cannot magically appear or disappear.
Key Point 3.15
Energy Conservation. In a closed system, energy is conserved; it can only be transformed between one form and another.- A closed system is one in which there is no mass or energy flow across its boundaries.
- It is believed that this conservation law is a
fundamental law of nature.
Key Point 3.16
Conservation of Mechanical Energy. In a closed system in which only conservative forces act, then
Ki+Ui=Kf+Ufwhere Ki, Kf, Ui and Uf are the initial and final kinetic and potential energies.
This law is useful because it tells us something that does not change, even when various types of energy are changing.
- The connections between kinetic energy, potential energy and energy conservation can be derived very elegantly.

Worked example
The system shown is released from rest with the 12kg block 3 m above the floor. Use the principle of conservation of energy to find the velocity with which the block strikes the floor. Neglect friction and inertia of the pulley. |
Solution
The total mechanical energy is conserved:
![\[ K_i + m_1gh^i_1 +m_2gh^i_2 = K_f + m_1gh^f_1 +m_2gh^f_2 \]](mastermathpng-1.png)




Hence
![\[ v=\sqrt{30}=5.48 ms^{-1} \]](mastermathpng-6.png)
[C] Violation of energy conservation?
- In some cases it may appear that energy conservation is violated. In fact this is not the case; what is happening is that energy is being transformed into a form that has not been accounted for e.g. heat, sound, electrical energy, strain energy, chemical energy. Often, this transformed energy is difficult to measure.
- One remedy is to reconsider what constitutes our ‘system’ in such a way as to account for the ‘lost’ energy. In doing this it is sometimes advantageous to introduce phenomological forces such as friction (S2.8) (which accounts for energy transformed into heat.)
- In assessing matters of energy conservation, one has to decide when and whether it is safe to ignore energy transformed into other hard-to-measure forms such as heat. In tutorial problems, you will often be given clues, or just told that energy is conserved.

Worked example
During a frisbee tournament, a 75g frisbee is thrown from a point 1.5 m above the ground with an initial speed of 15m/s. At some point in its flight it has a height of 2.25m and a speed of 12m/s
- Calculate the work done on the frisbee by its weight in reaching this point.
- Is mechanical energy conserved? If not, suggest a reason why.
Solution
- The frisbee rises 0.75m. Its weight is mg=0.075×10=0.75N directed downwards. Gravity is a conservative force so we can ignore the actual path taken and just use the vertical height change. The work done is then W=-mgΔh=-0.5625J
The initial mechanical energy is
The final mechanical energy is
Thus, for the Earth-frisbee system, mechanical energy is not conserved. The ‘loss’(=2.48J) is, of course, due to air resistance. Adopting instead the earth-frisbee-air system, we see that energy is conserved.
Learning Resources
![]() | HRW Chapter 8.5, 8.8 |
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