Base and Derived Quantities

• Physical quantities are classified as base (or fundamental) quantities and derived quantities. 7 base quantities are chosen to form the base units.

Base Quantity	Base Unit
mass	kilogram (kg)
length	metre (m)
time	second (s)
electric current	ampere (A)
temperature	kelvin (K)
amount of substance	mole (mol)
luminous intensity	candela (cd)

• Any other physical quantities can be derived from these base quantities. These are called derived quantities.

Prefixes

• Prefixes are attached to a unit when dealing with very large or very small numbers.

Power	Prefix
$10^{-12}$	pico (p)
$10^{-9}$	nano (n)
$10^{-6}$	micro ($\mu$)
$10^{-3}$	milli (m)
$10^{-2}$	centi (c)
$10^{-1}$	deci (d)
$10^3$	kilo (k)
$10^6$	mega (M)
$10^9$	giga (G)
$10^{12}$	tera (T)

Homogeneity of Units in an Equation

• A physical equation is said to be homogeneous if each of the terms, separated by plus, minus, equality or inequality signs has the same base units.

Uncertainty

• Absolute uncertainty of a measurement of $x$ can be written as $\Delta x$. This means that true value of the measurement is likely to lie in the range $x-\Delta x$ to $x + \Delta x$.
• Fractional uncertainty = $\dfrac{\Delta x}{x}$
• Percentage uncertainty = $\dfrac{\Delta x}{x}\times100%$
• If the values of two or more quantities such as $a$ and $b$ are measured and then these are combined to determine another quantity $Y$, the absolute or percentage uncertainty of $Y$ can be calculated as follows:
  • If $Y = a\pm b$, then  $\Delta Y = \Delta a+\Delta b$
  • If $Y = ab$ or $Y = \frac{a}{b}$ , then  $\frac{\Delta Y}{Y} =\frac{\Delta a}{a}+\frac{\Delta b}{b}$
  • If $Y = a^n$ then  $\frac{\Delta Y}{Y} = n\frac{\Delta a}{a}$

Errors

• Systematic errors are errors that, upon repeating the measurement under the same conditions, yield readings with error of same magnitude and sign.
• Random errors are errors that, upon repeating the measurement under the same conditions, yield readings with error of different magnitude and sign.

Accuracy and Precision

• The accuracy of an experiment is a measure of how close a measured value is to the true value. It is a measure of the correctness of the result.
• The precision of an experiment is a measure of how exact the result is without reference to what that the result means. It is a measure of how reproducible the results are, i.e. it is a measure of how small the uncertainty is.

Vectors

• A vector quantity has magnitude and direction.
• A scalar quantity has magnitude only.
• Addition of vectors in 2D: $\vec{a}+\vec{b}=\vec{c}$
• Subtraction of vectors in 2D: $\vec{a}-\vec{b}=\vec{d}$

• Methods of finding magnitudes of vectors:
  1. resolution of vectors into perpendicular components
  2. by scale drawing
  3. using: sine rule: $\frac{a}{\sin \alpha}=\frac{b}{\sin \beta}=\frac{c}{\sin \gamma}$ cosine rule: $a^2 = b^2 + c^2-2bc \cos \alpha$

A disc rotates clockwise about its centre O until point P has moved to point Q, such that OP equals the length of the straight line PQ. What is the angular displacement of OQ relative to OP?

A.   $\frac{\pi}{3}$ rad

B.   $\frac{2\pi}{3}$ rad

C.   $\frac{4\pi}{3}$ rad

D.   $\frac{5\pi}{3}$ rad

A siphon operates through the combined effects of gravity and air pressure, which work together to move liquid from a higher elevation to a lower one. Gravity is the primary force driving the flow, as it pulls the liquid from the higher container down through the siphon tube to the lower container. The liquid's potential energy, due to its elevated position, is converted into kinetic energy as it flows downward.

Air pressure plays a crucial supporting role by maintaining the continuous flow of liquid. Atmospheric pressure on the liquid's surface in the higher container pushes the liquid into the siphon tube. This pressure counteracts gravity's pull that might otherwise cause the liquid to fall back into the higher container. As the liquid moves downwards, it creates a partial vacuum in the upper part of the tube, allowing atmospheric pressure to push more liquid into the tube, sustaining the flow.

Thus, a siphon can continue to operate as long as the outlet is lower than the liquid surface in the source container, the tube remains filled with liquid, and atmospheric pressure supports the flow.

I find this video easy to understand and it may be useful for students to appreciate the wave property of matter and how it is observed via interference.  The video ends with a mind-boggling problem that when an attempt to detect the path of the electron, it goes back to behaving as a particle.

There's a whole series of "What the Bleep" videos that you might want to check out also. Be careful though, the rabbit hole is pretty deep.

Quoting from another website on what could have happened to each electron and to make the problem clearer (and hence more confusing):

The possibilities are: 1) the electron went through the left slit; or 2) the electron went through the right slit; or 3) the electron went through both slits. For the sake of logical rigor, we should add the possibility that 4) the electron went through neither slit (that is, it found some other way to get to the back wall). Now, one problem with possibility number 3 -- a single electron going through both slits -- is that, in nature, there is no such thing as half an electron. So if we found half an electron at both slits, we would have something really new; but that has to be a distinct possibility, considering that, in order to create the apparent interference pattern, something would have to radiate from both slits.

How are we going to find out? Well, we are going to put an electron detector at each slit. The electron detectors at the slits will be devices to keep watch over the passage through the slit. Every time an electron (or part of an electron) goes through, the detector will give a holler, "Hey, an electron (or part of an electron) just went through." In this way, we will be able to learn something about how the electrons get through the barrier in a double slit experiment.

As it turns out, when you put the electron detectors at the slits, the result is that the electron is always detected at one slit or the other slit. It is never found going through both slits. And it is never found going through neither slit. You send one electron through, you find it at one of the slits. We have eliminated possibilities number 3 (both slits) and number 4 (neither slit). The only results we find are possibilities number 1 (left slit) or number 2 (right slit), in equal proportions.

They call this phenomenon the measurement effect. When we measure something at the quantum level, the very act of measurement will have an effect on the thing itself.

This is a phenomenon that still has no classical explanation.

Even Richard Feynman called it "a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery."

This demonstration can be modified for use as a magic trick.

Materials:

1. Glass of water
2. Piece of cardboard that is larger than the mouth of the glass.

Procedure:

1. Fill the glass up with water.
2. Place the piece of cardboard over the mouth of the glass.
3. Holding the cardboard against the mouth of the glass, invert the glass.
4. Release the hand slowly.

Explanation

Water can remain in an inverted glass with the piece of cardboard underneath because atmospheric pressure is acting upward on the cardboard, holding it up together with the water. There is little air pressure within the g;ass, so the downward force acting on the cardboard is mainly the weight of the water, which is to the order of several newtons whereas atmospheric pressure exert an upward force of several thousand newtons.

Modification:

1. Drill a small hole in a plastic cup, near the base.
2. Seal the hole with your thumb and fill the cup with water.
3. Place the cardboard over the mouth of the cup.
4. Invert the cup together with the cardboard, while keeping your thumb over the hole.
5. Using a magic word as the cue, shift your thumb slightly to allow a little air into the cup. This will cause the cardboard and water to fall. As the air pressure within the cup is equal to that of the atmosphere.