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In addition to the revision notes for Pendulums. Energy in Simple Harmonic Motion on this page, you can also access the following Oscillations learning resources for Pendulums. Energy in Simple Harmonic Motion
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
|---|---|---|---|---|---|---|
| 10.2 | Pendulums. Energy in Simple Harmonic Motion |
In these revision notes for Pendulums. Energy in Simple Harmonic Motion, we cover the following key points:
Damped oscillations are a kind of SHM whose amplitude decreases with time by a factor of e - γ × t (the envelope). The equation of damped oscillations is
On the other hand, in sustainable SHM, the amplitude does not change with time. The envelope shows a horizontal function of the type x(t) = A0. Therefore, the equation of sustainable SHM is
Cosine function is used only for convention but it can be replaced by sine function as well by making the proper arrangements.
Sustainable SHM's are a special case of damped SHM's. They cannot exist without a continuous source of energy. Otherwise, oscillations will dump and eventually they fade away.
Any object or system moving in sustainable SHM possesses two kinds of energy: Kinetic Energy KE and Potential Energy PE. The equation for the mechanical energy of an oscillating spring in SHM is
The maximum values for potential and kinetic energy in an oscillating spring (when the other term of the ME formula is zero), are
and
There is another type of SHM besides the linear SHM. It is known as Angular Simple Harmonic Motion and its simplest case known as "simple pendulum".
A simple pendulum consists of a small mass m otherwise known as bob which is hung on a light and non-elastic string (thread) of length L attached to a fixed and rigid support.
The period T of oscillations in a spring doing SHM is
In a simple pendulum we substitute k = mg/L and thus, we obtain for the period of simple pendulum
The quantity
is known as the restoring force in a simple pendulum.
The equation of SHM for a simple pendulum considers the rotational parameter of angular displacement θ as dependent variable and the time t as independent one, i.e. it is an equation of type
where θ0 is the amplitude, i.e. the initial angle to the vertical direction, ω = 2π / T the angular frequency θ(t) is the angle to the vertical direction at a given instant t.
Using the derivation rules, we obtain
for angular velocity, and
for the angular acceleration in a simple pendulum.
Mechanical energy in a simple pendulum is
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