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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:

- What are damped and sustainable SHM and where do they differ from each other?
- How to calculate energy in SHM?
- What is simple pendulum?
- How to calculate the period of a simple pendulum?
- What are the factors affecting the period of a simple pendulum?
- Which are the parameters used to study the motion of a simple pendulum?
- How to find the angular displacement and angular velocity in a simple pendulum?

**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

x(t) = A_{0} × e^{ - γ × t} cos(ω × t + φ)

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) = A_{0}. Therefore, the equation of sustainable SHM is

x(t) = A_{0} × cos(ω × t + φ)

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

ME = KE + PE

=*m × A*^{2} × ω^{2} × sin^{2} (ω × t + φ)*/**2** + **k × A*^{2} × cos^{2}(ω × t + φ)*/**2*

=

The maximum values for potential and kinetic energy in an oscillating spring (when the other term of the ME formula is zero), are

PE_{max} = *k × A*^{2}*/**2*

and

KE_{max} = *m × A*^{2} × ω^{2}*/**2*

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

T = 2π × √*m**/**k*

In a simple pendulum we substitute k = mg/L and thus, we obtain for the period of simple pendulum

T = 2π × √*L**/**g*

The quantity

F = -m × g sinθ

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

θ(t) = θ_{0} × cos(ω × t)

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

ω(t) = *dθ**/**dt** = θ*_{0} × ω × sin(ω × t)

for angular velocity, and

α(t) = *dv**/**dt*

= -θ_{0} × ω^{2} × cos(ω × t)

= -ω^{2} × θ(t)

= -θ

= -ω

for the angular acceleration in a simple pendulum.

Mechanical energy in a simple pendulum is

ME = GPE + KE

= m × g × L (1 - cos θ) + m ×*v*^{2}*/**2*

= m × g × L (1 - cos θ) + m ×

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