Energy and Damping In SHM
Previously, the Velocity of a particle undergoing SHM was found by :-
If this formula is combined with the Kinetic Energy formula, a method to find the Kinetic Energy of the particle at any given point in its motion can be derived :-
As this is being modelled as an ideal system with no Energy loss :-
Therefore at the equilibrium point of the oscillation, Etot = Ek + 0 it can be shown that :-
By substituting into the general equation above, the Potential Energy can be shown by :-
The graph below shows how the Ek and Ep vary across the entire oscillation :-
Example 1 -
The following graph shown the Potential Energy of a particle with mass = 0.4 kg undergoing SHM
Find the value of the maximum Potential Energy for the system
Calculate the Force constant k for the oscillating system ( k has units Nm-1 )
Find the amplitude at which the Ep = Ek
From graph, Ep is at maximum when y = A, giving max Ep = 0.1 J
Ep = 0.5 k y2
0.1 = 0.5 k ( 0.05 )2
k = 80 Nm-1
Ep = Ek
SHM and Damping
So far, all the work on SHM has assumed that there is no energy lost in the oscillation, but this is not realistic.
In order to understand how Simple Harmonic Motion occurs in realistic situations, and understanding of Damping is required.
There are three types of Damped systems that can applied to Simple Harmonic Motion :-
1. Under Damped
2. Critically Damped
3. Over Damped
All classically oscillating systems will show damping as Energy is lost from the system to the environment, usually in the form of frictionally generated heat.
Under-Damped System
In an under-damped system, the object will oscillate back and forth around an equilibrium point, losing Energy each cycle until it comes to rest at equilibrium.
Examples of under-damped systems includes the pendulum and spring experiments from previous lessons on SHM.
Under-damped systems can be further catagorised as :-
1. Lightly Damped - a pendulum takes a long time to come to rest as the air resistance is small.
2. Heavily Damped - a hydrometer 'bobbing' up and down in a liquid comes to rest quickly due to the large frictional Forces with the liquid.
The graph below shows the motion of an under-damped object undergoing SHM :-
Over Damped System
In an over-damped system, the object experiences a large resistance to motion, taking a long time to reach the equilibrium position (if at all).
The graph below shows the motion of an over-damped object undergoing SHM :-
Critically Damped System
In a critically damped system, there is a frictional Force which is just enough to bring the oscillating object to rest at equilibrium quickly.
The graph below shows the motion of a critically damped object undergoing SHM :-
