Category: 08 Simple Harmonic Motion

# 8.B More Demonstrations

These demonstrations are arguably beyond the H2 syllabus. Nevertheless, they are fun and they may help you understand the H2 syllabus.

Bartonโs Pendulum

The Quarter-cycle Lag between Driver-Driven

The Tuning Fork Resonance

The Coupled Pendulum

The Coupled Inverted Pendulum

The Tacoma Narrows Bridge

# 8.A Angular Frequency vs Angular Velocity

These 17 oscillations are arranged (from left to right) according to their angular frequency ฯ in ascending order. See the resulting pattern!

Here is a physical demonstration using pendulum of increasing length.

The supposed link between angular velocity (circular motion) and angular frequency (of oscillation).

# 8.5.2 Effects of Damping on Resonance

How does damping affect the amplitude of forced oscillations? Here is a demonstration.

Here is an explanation.

And here is an applet.

# 8.4.1 Damped Oscillations Applications

Suspension systems are designed to be critically damped.

Skyscrappers are inverted pendulums that must be damped.

# 8.3.2 SHM Energy in x-Domain

Here is a video showing you how to derive the energy-displacement formula at the snap of the finger!

Animation of the energy-displacement graphs.

Remember that KE+PE=constant=TE

# 8.3.1 SHM energies in time-domain

Remember this: there are TWO energy cycles in each oscillation.

Animation of the KE-time graph.

Animation of the PE-time Graph

Animation of the TE-time graph

# 8.3 Energy of SHM

Work done by Restoring Force increases KE.

Work done against Restoring Force increases PE.

# 8.2.1 Natural Frequency

Every pendulum is “born” with a natural frequency. This frequency (and period) is not affected by the amplitude of oscillation.

The natural frequency of a spring-mass system depend on the stiffness (the spring constant of the spring) and the inertia (the mass of the mass) of the system.

The formula for the spring-mass system can be derived in just a few lines from N2L.

The natural frequency of a pendulum is dependent only on its length. (The acceleration of free fall is the other factor, but we usually cannot choose the value of g.)

Very intricate patterns can be traced out by a “double pendulum”. See the pendulum showing off its artistic streak below.

The CRO (cathode ray oscilloscope) goes artistic too.

# 8.2 Restoring Force

The restoring force keeps the mass accelerating towards the equilibrium position. The inertia (of the mass) causes the mass to always overshoot the equilibrium position. Hence the eternal karma of oscillations.