Introduction of classical mechanics


Classical mechanics is all about describing the motion of macroscopic solids and fluids under the influence of forces. For example how planets move around the Sun why apples fall down on people's heads and how planes can fly around and many more things that we're going to talk about in this course. Now, who invented all of this well some of the first people thinking about classical mechanics were the Greeks but unfortunately they were not very good at it so most of their ideas are irrelevant for today. Physics later Galileo Galilei came up with some cool scientific ideas around 1600 after his death. The famous Isaac Newton came up with calculus and laws that describe the motion of macroscopic objects which are still in use today. A few years later Joseph Louis Lagrange and William Hamilton developed different formulations of Newton's laws to solve more complex problems a lot easier.

Whats is its boundaries?


Now, what does classical mechanics not able to describe well. You can't describe objects moving close to the speed of light that's where Einstein's theory of relativity comes into play. You're also not able to describe tiny objects with noticeable Debry wavelengths. Here the quantum theory is needed and I might also do a website serious about this in the future. As you can see classical mechanics is not a 100% accurate and complete theory but it's still very useful within its boundaries.

What are we going to talk about?





Well at first we're going to talk about

  1. Space and time and coordinate systems then we'll talk about

  2. Kinematics which is all about how stuff moves around after that I'll introduce

  3. Newton's laws of motion which described why stuff moves around and after that, we'll talk about

  4. Ballistics which you can think of as throwing stuff around on short distances then we'll talk about

  5. Momentum and collisions the next topic will be

  6. Work and energy then we'll talk about oscillations in different circumstances and the Fourier series

  7. Wave mechanics but I'll go deeper into this topic at a later point in this course and then I'll introduce

  8. Lagrangian and Hamiltonian mechanics after that we're ready to talk about the

  9. The two-body problem we'll talk about

  10. Accelerated frames of reference rigid bodies will be the next topic and

  11. Here we will especially be talking about the rotation of rigid bodies


Limitation or failure of classical mechanics



  1. Black body radiation

  2. Photoelectric effect

  3. The hydrogen atom


Black body radiation


Any object with a temperature above absolute zero emits light at all wavelengths. If the object is completely black (so it does not reflect any light), then the light emitted from it is called black body radiation.

Blackbody radiation energy is not evenly distributed across all wavelengths of light. The spectrum of black body radiation (below) shows that some wavelengths have more energy than others. Three, three spectra for different temperatures are shown.

Here are some experimental facts about black body radiation:

a. The black bed spectrum depends only on the temperature of the object, not on it. An iron horse, a ceramic vase, and a piece of charcoal --- all emit the same black bed spectrum if they have the same temperature.

b. As the temperature of an object rises, it emits more energy than the black body in each wavelength.

c As the temperature of an object rises, the peak wavelength of the blackbed spectrum decreases (crystal). For example, blue stars are hotter than red stars.

d. The black body spectrum is always short on the left hand side (short wavelength, high frequency side).

Description of Classical Physics: Light is an electromagnetic wave that is generated when an electric charge vibrates. (Strictly speaking, "vibration" refers to how charge moves --- fast, slow, or changes direction.) Now remember that heat is just the dynamic energy of random motion. In a hot object, electrons vibrate in random directions and produce light as a result. A hotter object means more vibrational vibrations and more light is emitted by a hotter object --- it is brighter. So far, so good. But classical physics could not explain the shape of the black bed spectrum.

In a hot object, electrons can vibrate with a range of frequencies, ranging from very small to very large number of vibrations per second. In fact, there is no limit to how much frequency can be. Classical physics said that every frequency of vibration should have the same energy. Since there is no limit to the range of frequency, there is no limit to the energy of electrons moving at high frequencies. This means that according to classical physics, there should be no limit to the energy of the moving light of electrons at high frequencies. Wrong !! Experimentally, the black bed spectrum is always smaller on the left hand side (short wavelength, high frequency)

Around 1900, Max Planck came up with the solution. He suggested that the classic idea was that each frequency of vibration should have the same energy. Instead, he said, energy is not evenly shared by electrons vibrating at different frequencies. "Energy comes in handy," Planck said. He called the collision of energy quantum. The size of a collision of energy --- a quantum --- depends on the frequency of the vibration. Planck's law for the amount of vibrating electrons is:

Quantum energy = (a calibration constant) x (frequency of vibration)

Or

E = hf

Where H, the calibration constant, is today called Planck's constant. Its price is about 6 x 10-34, very small!

So how does this explain the spectrum of black body radiation? Planck said that an electron rotating with a frequency F can receive only 1 HF, 2 HF, 3 HF, 4 HF, ... energy. That is,

Energy of moving electrons = (any number) x hf

But if the electron is going to vibrate, then at least a certain amount of energy must be kept. If it does not have at least 1hf of energy, it will not vibrate at all and will not produce any light. "A ha!" Planck said: A quantum at high frequency, the amount of energy in HF is so high that high frequency vibrations can never go! That is why the black body spectrum is always smaller on the left side (higher frequency).

Photoelectric effect


When light shines on the surface of a metallic substance, the electrons in the metal absorb the energy of light and they can escape from the surface of the metal. This is called the photoelectric effect, and is used to generate electric power that drives many solar powered devices. Using the idea that light is a wave with energy evenly distributed throughout the wave, classical physicists expected that when using extremely dim light, enough light to emit electrons from a metal surface It will take some time for energy. Wrong !! Experiments have shown that if electrons can be extracted from a metal by a certain frequency of light, it does not matter how dim the light is. There is never a delay.

In 1905, Albert Einstein came up with a solution. If Max Planck's idea that energy comes in quantas is correct, then light must contain a stream of energy waste. Einstein said that each part of the energy of light is called a photon, and each photon has an energy equal to HF (constant frequency of light is the constant time of the plank). Therefore the energy of light is not evenly distributed with the wave, but is concentrated in the photons. Dim light means fewer photons, but simply changing the light (without changing its frequency) does not change the energy of an individual photon. Therefore, for a certain frequency of light, if a single photon has enough energy to expel electrons from the metal surface, the electrons will be expelled only after the light is activated and the electrons hit the metal.

Hydrogen atom


When a small tube of hydrogen gas is heated, the light begins to glow and begin to emit. Unlike black body radiation emitted from hot dense solids or gases, this light has only a few colors (wavelengths): one red wavelength, one turquoise and several violet. At the turn of the century, classical physicists thought that they must be able to understand hydrogen, because it is the simplest atom. The hydrogen nucleus consists of a positively charged proton, around which a negatively charged electron rotates. The electric gravity between a positive proton and a negative electron keeps the electron in orbit, just as the gravitational pull between the sun and the earth keeps the earth in orbit. There was only one problem. Classical physics says that because the rotating electron is constantly changing direction, it must emit electromagnetic radiation --- light. As a result, electrons lose energy permanently. In fact, physicists calculated that an electron should lose all its energy and orbit a proton in just 0.000000000001 seconds! In other words, the atom should not be present for more than 10-12 seconds. Wrong !!

Niels Bohr provided an explanation in 1913. In the Bohr model of the hydrogen atom, electrons cannot orbit protons in any size. There are only certain orbits, and each orbit has a certain radius and a certain energy. Bohr invented a rule by which he could calculate the size and energy of each orbit. If you want to know, Bohr's principle says this

2Ï€ x (electron mass) x (speed of electron orbit) x (radius of orbit) = (any number) x h

To say the least, which is not even more obvious! (Numbers will be 1 for the smallest orbit, 2 for the next orbit, and so on.) Bohr also devised a new rule to define the stability of a hydrogen atom --- because it takes more than 0.000000000001 seconds. Why can it last? When an electron is in orbit, the electron does not emit electromagnetic radiation, he said. Bohr did not explain why, he just proposed a new law of nature. And nature agreed with Nels Bohr. Its new model of hydrogen lengthened the waves of hydrogen gas, which was exactly what was being measured.

Question: If electrons do not produce light when they are in stable orbit, then where is the light source from hydrogen? Answer: According to Bohr, electrons have more energy when they are in large orbits. If an electron falls below a large orbit into a small orbit, it loses energy. According to the law of conservation of energy, the energy lost from electrons must go somewhere. Bohr explained that a photon carries lost energy from a hydrogen atom. That is,

Photon energy = (electron energy in large orbit) - (electron energy in small orbit)

It works the other way around. If a photon attacks an atom, the atom can absorb the photon and its energy if (and only if) the energy of the photon is equal to the difference between the two orbital energies. In this case, an electron uses photon energy to jump from small orbit to large orbit. This is called quantum jump.

Types


In our investigation of classical mechanics we will study several different types of motion, including:

Interpretive movement - The movement by which the body moves from space to place (such as the movement of a bullet from a gun).
Rotational motion - a movement by which the extended body changes orientation, with respect to other bodies in space, without changing position (e.g., the movement of the top of the wheel).
Oscillatory Movement - A movement that repeats itself over a period of time (such as a grandfather's clockwise motion).
Circular Motion - A movement by which the body moves in a circular orbit about another fixed body [e.g., the (approximate) motion of the earth about the sun].