# Pendulums. Energy in Simple Harmonic Motion

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10.2Pendulums. 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?

## Pendulums. Energy in Simple Harmonic Motion Revision Notes

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) = A0 × 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) = A0. Therefore, the equation of sustainable SHM is

x(t) = A0 × 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 × A2 × ω2 × sin2 (ω × t + φ)/2 + k × A2 × cos2⁡(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

PEmax = k × A2/2

and

KEmax = m × A2 × ω2/2

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 sin⁡θ

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) = θ0 × 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) = /dt = θ0 × ω × sin⁡(ω × t)

for angular velocity, and

α(t) = dv/dt
= -θ0 × ω2 × cos⁡(ω × t)
= -ω2 × θ(t)

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 × v2/2

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