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Boiling

Specific Heat Capacity

In fact, water has the highest specific heat capacity (4.2 kJ kg^-1 K^-1) of any liquid. It takes a lot of heat to change the temperature of water. This is the reason the temperature of a pond or sea water does not vary much over the course of a day. Imagine what happens to the frogs and fish if water has the same specific heat capacity as asphalt or cement. Haha.

Appendix B: Natural Frequency of Simple Pendulum (Beyond Syllabus)

To derive the pendulum’s period of oscillation, it is best to analyze it as a rotational motion.

We will still be applying N2L, but Instead of force F, inertia m and acceleration a, we have to work with torque $\displaystyle \displaystyle \tau$ , moment of inertia I (a body’s resistance to rotate, which for the pendulum is mL²) and angular acceleration $\displaystyle \displaystyle \ddot{\theta }$ .

$\displaystyle \displaystyle \begin{aligned}{{\tau }_{{net}}}&=\mathrm{I}\ddot{\theta }\\mg\sin \theta \times L&=m{{L}^{2}}\ddot{\theta }\\\ddot{\theta }&=\frac{{g\sin \theta }}{L}\end{aligned}$

If θ is small, we canmake the $\displaystyle \sin \theta \approx \theta$ approximation. (Yes. This means that our formula is valid only for small amplitude oscillations)

$\displaystyle \displaystyle \ddot{\theta }=\frac{g}{L}\theta$

Comparing this with the SHM equation $\displaystyle a=-{{\omega }^{2}}x$ ( $\displaystyle \displaystyle \ddot{\theta }=-{{\omega }^{2}}\theta$ ), we can deduce that

$\displaystyle {{\omega }^{2}}=\frac{g}{L}$

Hence, the natural frequency is $\displaystyle {{f}_{n}}=\frac{1}{{2\pi }}\sqrt{{\frac{g}{L}}}$ and the natural period is $\displaystyle {{T}_{n}}=2\pi \sqrt{{\frac{L}{g}}}$ .

From $\displaystyle \displaystyle {{\omega }^{2}}=\frac{{\ddot{\theta }}}{\theta }$ , we can see that the natural frequency depends on the (angular) acceleration per unit (angular) displacement ratio. A higher g increases the restoring force (torque), whereas a shorter L decreases the (moment of) inertia, both resulting in a higher (angular) acceleration per unit (angular) displacement.

The mass of the pendulum, surprisingly, does not affect the period. This is because the restoring torque $\displaystyle (mg\sin \theta )$ and moment of inertia (mL²) are both proportional to m. So m cancels itself out when it comes to angular acceleration. (This is similar to how all object free fall at g regardless of mass).

Appendix A: Angular Frequency vs Angular Velocity

The angular frequency is a rather abstract concept which deserves some discussion.

$\displaystyle \omega =\frac{{2\pi }}{T}=2\pi f$

At the most basic level, from $\displaystyle \omega =2\pi f$ , you should appreciate that the angular frequency ω is simply the frequency of the oscillation multiplied by $\displaystyle 2\pi$ .

At a more abstract level, from $\displaystyle x={{x}_{0}}\sin \omega t$ , we can think of $\displaystyle \omega t=\theta$ as the phase angle of the oscillation. So ω is the “velocity” at which the phase of the oscillation progresses.

The angular frequency ω in SHM is actually very similar in concept to the angular velocity ω in circular motion. (This explains why they are both given the symbol ω.)

For oscillations, $\displaystyle \omega =\frac{{d\theta }}{{dt}}=\frac{{2\pi }}{T}$ is the rate of change of phase angle.

For circular motion, $\displaystyle \omega =\frac{{d\theta }}{{dt}}=\frac{{2\pi }}{T}$ is the rate of change of angular displacement.

Lastly, a circular motion collapsed into one dimension is actually an SHM. For example, the displacement in the y-direction of the circular motion illustrated below is actually an SHM. The circular motion’s radius R, is the SHM’s amplitude R. The angular velocity ω of the circular motion is also the angular frequency ω of the SHM.

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Animation

Circular Motion vs Oscillation

Pendulum Wave

Demonstration

Pendulum Wave (Harvard Natural Sciences)

8.5.3 Effect of Damping on Resonance

In a forced oscillation, the final amplitude reached is the amplitude at which the rate of gain of energy (from the driver) is matched by the rate of loss of energy (to the surrounding). It is kind of analogues to how the final terminal velocity is the velocity at which the weight is matched by the drag force.

Light damping is of course the reason for an oscillation to lose energy (to the surrounding). The resonance amplitude is thus dependent on the amount of light damping. (It is meaningless to talk about resonance for systems with heavy damping because there is no oscillation to talk about)

The resonance graphs under different amounts of light damping are shown below.

Notice that

damping causes the entire resonance curve to be lower, not just at the resonance frequency, but at every frequency (except $\displaystyle f=0$ ).
In theory, at zero damping, the resonance amplitude can reach infinity.
Damping causes resonance to occur at a frequency slightly lower than the oscillator’s natural frequency. So the resonance peaks will shift towards lower frequency as damping increases. But the shift is not significant under very light damping conditions.

Nearer to critical damping, the shift in the resonance frequency becomes obvious. But by then the resonance amplitude is so low it is not much of a resonance phenomenon.

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Demonstration

SHM with Damping

Video Explanation

Effects of Damping on Resonance

Concept Test

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8.5.2 Resonance Curve

Consider an oscillatory system with natural frequency f₀. Let’s couple this system to a driver. We can set the driver to vibrate with a fixed amplitude A₀ at different driving frequency f.

If we plot on a graph how the amplitude of the forced oscillation A varies with the driver’s frequency f, we obtain the so-called resonance curve.

Notice that

Amplitude A peaks at $\displaystyle f={{f}_{0}}$ .

When the driver is driving at the natural frequency of the oscillator, it’s like a match made in heaven. The energy transfer from the driver to the oscillator is at its most efficient. In fact, the oscillator is able to accumulate the energy transferred and reaches a maximum amplitude that is many times larger than the amplitude of the driver.

As f approaches 0, A approaches A₀.

When a slow driver meets a quick oscillator, the oscillator is so “nimble” it tracks the motion of the driver exactly. So the oscillator has the same amplitude of oscillation as the driver.

As f approaches infinity, A approaches zero.

When a quick driver meets a slow oscillator, the oscillator is so “retarded” it simply cannot respond in time. The driving forces changes too rapidly for the oscillator and its amplitude is stuck at zero.

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Video Explanation

Resonance Curve

Concept Test

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8.5.1 Forced Oscillation

There are two ways to get an oscillation going. The first way is to displace the oscillator from its equilibrium position and then release it. The oscillator will oscillate at its natural frequency. This is called a free oscillation. The second way is to exert a small but continuous external periodic driving force to the oscillator. In this case, the oscillator will be forced to oscillate at the frequency of the driving force. This is called a forced oscillation.

For a free oscillation, the energy is transferred to the oscillator in “one lump sum” at the beginning when the oscillator was displaced. The amplitude of oscillation starts off at a maximum and decreases gradually over time due to damping.

For a forced oscillation, however, energy is continuously transferred to the oscillator by the periodic driving force. The amplitude of the forced oscillation starts at zero, and increases over time, and stabilizes at a final amplitude when the rate of input of energy from the driver is matched by the rate of loss of energy to the surrounding due to damping.

The efficacy of energy transfer from the driver to the oscillator depends on how close the driving frequency is to the natural frequency of the oscillatory system. When there is a large mismatch, the energy transfer from the driver to the oscillator is inefficient, and the forced oscillation will only attain a small amplitude.

Resonance occurs when the driving frequency matches the natural frequency of the oscillator. At resonance, transfer of energy from the driver to the oscillator is at its most efficient, allowing for maximum amplitude to be attained.

Many teachers are very fond of the swing analogy, that you must push the swing at the correct timing if you want the swing to go higher and higher. Obviously, if you push the swing when it is coming back towards you, you’re being destructive to the amplitude building process. So in a similar (but not exact) manner, to achieve resonance the periodic driving force must be “synchronized” to the natural frequency of the oscillator, to ensure only positive (and no negative) work is done all the time to the oscillator.

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Demonstrations

Free Oscillation vs Forced Oscillation

Resonance

Barton’s Pendulum

Fortune Telling Sticks

Tuning Fork Resonance

Tacoma Narrows Bridge

Millennium Bridge

Applet

Final-State Amplitude (ngsir)

Concept Test

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Interesting

Quarter Cycle Lag

Coupled Pendulum

Coupled Inverted Pendulum

Metronome Synchronization

8.4.2 Practical Applications of Damping

There are many oscillatory systems around us in our daily lives. Some of these oscillations are nuisances and should be stopped as quickly as possible.

For example, vehicles are actually inverted spring-mass systems. The body of a car (mass) sits on top of its suspension system (spring). Every time the car hits a bump, or drives over an uneven surface, the car goes into oscillation. Shock absorbers are designed to be critically damped so that the car returns to the equilibrium position in the shortest amount of time.

Buildings are also inverted pendulums. They may look rigid to our eyes, but they are actually elastic and bendable. They too can be set into dangerous oscillation by strong gusts of wind, or ground tremors. For this reason, skyscrapers are equipped with devices to damp these oscillations.

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Demonstrations

Suspensions and Skyscrapers

8.4.1 Light, Heavy and Critical Damping

If an oscillation is completely undamped, the oscillation will go on forever. In practice however, all oscillations must encounter some damping forces (e.g. air resistance, friction). Damping saps the oscillation’s energy continuously, causing its amplitude to decrease continuously over time.

The behavior of damped oscillations has been studied extensively, in particular for the case of the damping force being proportional to the velocity. It turns out that there are three distinct outcomes depending on the degree of damping.

Light Damping

If the oscillation is subjected to light damping (a.k.a. underdamping), the amplitude of the oscillation decreases gradually over time. The displacement-time graph is roughly enveloped by an exponential decay function. The higher the amount of damping, the faster the rate of decay and the steeper the slope of the exponential decay function.

Lengthening of the Period

Damping causes an oscillation to have a longer period than if it were undamped. However, the lengthening of the period under very light damping is very small. In fact, the lengthening of period is more often than not neglected in the H2 syllabus.

Heavy Damping

If the oscillation is subjected to heavy damping (a.k.a. overdamping), the oscillator returns slowly to the equilibrium position without ever going past the equilibrium position. In that sense, heavy damping totally completely prevented any oscillation.

Critical Damping

Critical damping is the transition point between light and heavy damping. Critical damping causes a displaced oscillator to return to the equilibrium position in the shortest amount of time and without crossing the equilibrium position. Because of this, critical damping is the design objective in many engineering applications (e.g. shock absorbers, car suspension systems etc.).

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Demonstration

Damping in Water

Critical Damping using Magnetic Braking

Animation

Light, Heavy and Critical Damping (ngsir)

Slides

x-t Graph under Increasing Damping

8.3.3 SHM Energy in the x-domain

From $\displaystyle v=\pm \omega \sqrt{{{{x}_{0}}^{2}-{{x}^{2}}}}$ , we can write $\displaystyle KE=\frac{1}{2}m{{v}^{2}}$ as

$\displaystyle \displaystyle KE=\frac{1}{2}m{{\omega }^{2}}({{x}_{0}}^{2}-{{x}^{2}})$

We can then write PE as

$\displaystyle \displaystyle \begin{aligned}PE&=TE-KE\\&=\frac{1}{2}m{{\omega }^{2}}{{x}_{0}}^{2}-\frac{1}{2}m{{\omega }^{2}}({{x}_{0}}^{2}-{{x}^{2}})\\&=\frac{1}{2}m{{\omega }^{2}}{{x}^{2}}\end{aligned}$

A few things to note.

Both KE and PE are quadratic graphs and are mirror image of each other
The graphs intersect at $\displaystyle E=\frac{1}{2}TE$ , $\displaystyle x=\pm \frac{{{{x}_{0}}}}{{\sqrt{2}}}$ .

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Video Explanation

Energy-Displacement Graphs of SHM

Concept Test

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