S6.5 Energy conservation in SHM
[A] Why SHM entails energy conservation
- In section S6.2 we saw that
SHM emerges when we have a mechanical system with an associated potential
energy:
- The minimum of the PE identifies an equilibrium position.
- SHM results if the system is displaced a little from that point, provided the equilibrium is stable.
- If the only force that affects the motion is the one associated with this potential energy then the system is conservative.
- In section S3.7 we saw that the total (kinetic and potential) energy in such a system is conserved (ie constant).
- We must thus expect that, quite generally, mechanical energy is conserved within SHM.
- We show this explicitly for the mass-spring system.
Analysis
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Analysis
- Consider the mass-spring system once more.
- The potential energy is controlled by
the displacement x:
- The kinetic energy is controlled by
the velocity
:
- From our solution to the general SHM equation
The PE and KE at time t are then
and- From Key Point 6.4 we have
- Thus the total energy at instant t is
- We may conclude as follows:
Key Point 6.7
In a mechanical system exhibiting SHM the sum of the kinetic and potential energy remains constant at its initial value:![\[ K + U = E = \frac{1}{2} m x_m^2 \omega^2 =\frac{1}{2} k x_m^2 \]](mastermathpng-8.png)

Commentary
- Note that E is a constant which does depend on the initial conditions.
- We can set the system going with any energy we choose by adjusting the initial position, or the initial speed.
- The total energy can be recognised as the maximum kinetic
energy (or maximum potential energy)
the system has in its oscillation cycle:
[B] Variation of K and U during the SHM cycle
- Although the sum of K and U remains constant during the SHM cycle each of them varies individually.
- One can think of energy being continually exchanged between the two forms.
- We can display the variation as a function of time or as a function of space (displacement).
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- The ’distance’ between the x-axis and the parabola is the potential energy
- The ’distance’ between the parabola and the total energy is the kinetic energy.
- The particle never gets beyond ±xm because its kinetic energy runs out there.
- This turns out to be a useful platform for thinking about the quantum world …
[C] From energy conservation to the SHM equation
- We have shown that a system exhibiting SHM displays conservation of mechanical energy.
- We can reverse this argument:

Analysis
Suppose we have a system whose potential and kinetic energies can be written in the form
where S and I are constants, while (for this argument) it will help to use v for
.
- The mass spring system is a special case of this general form
- This time we assume energy conservation
so that
- Now differentiate this result with respect to time.
We need to use the chain rule:
andwhere a is the acceleration.
- Now the statement E is constant is equivalent to
- Combining these results gives:
- Cancelling common factors and setting
gives
which is the SHM equation. - We conclude the following:
Key Point 6.8
If the total energy of a system can be written as![\[ E = U + K = \frac{1}{2} S x^2 +\frac{1}{2} I \dot{x}^2 =\mbox{ constant} \]](mastermathpng-19.png)
![\[ \omega= \sqrt{\frac{S}{I}} \]](mastermathpng-20.png)

Mechanics without forces?
Mechanics without forces seems a contradiction in terms. But classical mechanics can be formulated entirely without forces. These formulations (known as Lagrangian Mechanics and Hamiltionian Mechanics) provide the most appropriate techniques for dealing with harder problems. Some of you will meet these methods in later years. One nice thing to look forward to: they allow you to dispense with vectors!Learning Resources
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