Resonant Frequencies

v_n = nth harmonic
n = integer
v₀ = fundamental frequency

We’ve all seen those opera singer references which claim that at a particular frequency, the singer’s voice can break a wine glass. The physics behind the trick is that of resonance.
Basically, anything that can vibrate has some fundamental frequency of vibration which depends on its physical properties. A fixed, vibrating string is the best way to visualize this. If you pluck it at just the right point, it starts to vibrate. The string vibrates in a manner that the fixed ends and the point you plucked(approximately) stay at their position while the rest of the string oscillates up and down. If it is one of those resonant frequencies, the vibrations will last for some time(ideally they should last forever). Otherwise they’ll dampen quickly.
The string prefers to vibrate at these resonant frequencies, the lowest of which is called the fundamental frequency. Every object has a fundamental frequency. If you put another object near it vibrating at one of the resonating frequencies, the first object will pick up the vibration as well. This is called resonance (e.g. a tuning fork and a pipe).

More info:
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/funhar.html#c1

Some examples:

Difference between harmonics and overtone:

A beautiful experiment with salt:

For more equations in Physics, see Famous equations in Physics

D’Alembert’s Principle

A virtual displacement of a system refers to a change in the configuration of the system as the result of any arbitrary infinitesimal change of the coordinates δr_i, consistent with the forces and constraints imposed on the system at the given instant t. (from Goldstein)

The principle of virtual work states that

If the net virtual work (during any virtual displacement δr) done by all external forces on a rigid body consistent with the constraints imposed on the body, is zero then the body is in equilibrium.

However, this principle requires that internal frictional forces do no work during the virtual displacement (all other types of internal forces cancel each other).

                                                       δW= ∑ F_i . δr_i = 0                                        (summed over all particles)

The principle above considers only static constraints on the system, when the constraint forces do no work at all and only hold the system in place. However, in the dynamics problems, we can extend the principle above to include the forces of constraints as well, by expressing them as inertial forces.

In this case, the net work on the system changes to the sum of the work done by the external (or applied) forces and work done by inertial forces (describing constraints):

                                                       δW= ∑ F_i . δr_i +∑ F_i^*. δr_i = 0                    (summed over all particles)

Or, equivalently,

The equation above summarizes d’Alembert’s principle which states that

the sum of the differences between the forces acting on a system of mass particles and the time derivatives of the momenta of the system itself along any virtual displacement consistent with the constraints of the system, is zero.

Wikipedia

This principle is somewhat similar to the Newton’s second law of motion. However, it is not that easy to prove that they are equivalent. Anyhow, this is not very useful in the present form. We need to derive equations of motion from it in order to apply it in real life situations.

Particle Physics: Particles and the Standard Model

Particles are everywhere

We know that matter consists of atoms which in turn consists of neutrons, protons and electrons. There are also photons, that carry energy associated with the electromagnetic interaction. The electron appears to be a fundamental particle since we never see it decay into something else. However, the neutrons and protons seem to have some internal structure. A neutron, if left on its own, can disintegrate^[1] into a proton and an electron in nearly 15 minutes:

$n \rightarrow p + e^{-} + \bar{\upsilon }_{e}$

The particle is called anti-neutrino. It is the antiparticle of neutrinos (more on this later) that are presumably massless and uncharged point-like particles which appear mostly in processes involving electrons (or antielectrons). The electrons and neutrinos do not appear to have any internal structure (unlike protons and neutrons) and are thus classified as some of the “elementary particles”.

Experiments such as the Large Hadron Collider and studies of cosmic rays have resulted in the discovery of even more elementary particles. They were named pions, muons, kaons, tau etc. until we ran out of greek letters! All of these particles present a challenge to the physicists because many of them exist for miniscule durations and at high energies which are not easily attainable. However based on what we know (and we know quite a bit), there are certain patterns in their properties and behaviour. These patterns make them easier to sort into categories.

The standard Model

The standard model is widely accepted as the most fundamental level of classification of elementary particles we have yet achieved. This model assumes that at the heart of everything there are certain elementary particles which can interact with each other according to certain conservation laws. There are basically two kinds of particles, leptons and quarks (collectively known as the Fermions). The interactions among them (forces) are modeled by carrier particles, known as Bosons¹.

Thus, the standard model divides elementary particles into two categories, Bosons (e.g. photons, W and Z bosons etc.) and Fermions (electrons, quarks etc.).The Fermions are further categorized into Leptons and Quarks.

It also lists their properties and interactions. The interactions are governed by Conservation Laws. In this way, the standard model is a theory in itself^[2]. Although there are certain anomalies in this model but for the sake of our discussion we assume it is correct.

You can check out particleadventure.org for details. I love this website and they have an app too. Happy learning!

Unlike Newtonian Physics, we do not treat forces as anything different from particles. Essentially all particles are considered to be the manifestation of some field. Photons are thus, equivalent to electromagnetic fields and particles have mass due to Higgs field.

Disclaimer: All of the information provided above is a simplification. Most of these topics require understanding of advanced concepts and involve mathematical difficulties in explaining them theoretically. I recommend you refer to a standard textbook such as Introduction to Elementary Particles by David Griffiths for clear and precise explanations.