Circular motion

The centripetal force is the resultant force acting towards the centre that makes a body travel in a circle:

\(F=m \omega^2 r={mv^2 \over r}\)

Vectors

Vectors are usually drawn with an arrow in the correct direction, where the length of arrow represents the magnitude. A force is an example of a vector.

Balancing forces

If the force vectors acting on a body add up to zero then we say they are balanced. To determine if this is the case, resolve the forces into two perpendicular directions (e.g. vertically and horizontally) and see if they cancel out in each.

Trigonometry

A trigonometric identity is an equation that applies for any value of angle. The identity that you must learn is:  \(\tan \theta={\sin \theta\over \cos\theta}\)

Energy transfers

Kinetic energy (E_k) is due to movement, equal to the amount of work required to make a body move.

\(E_k={1\over 2} mv^2\)

Potential energy (E_p) is positional, equal to the amount of work done in taking a body from a place of zero potential energy to the place in question.

\(E_p=mgh\)

Inelastic collisions

When bodies stick together, E_k isn't conserved. Energy is lost. However, momentum is conserved.

Ideal gases

The ideal gas equation relates pressure, temperature, volume and the amount of gas:

\(pV=nRT\)

To calculate the internal energy, U, of a gas we assume that potential energy is zero: 

\(U=\sum E_p+\sum E_k=\sum E_k\)

\(\Rightarrow U=N{3\over 2} kT\)

Gas laws

Boyle's law: The pressure of a fixed mass of gas at constant temperature is inversely proportional to its volume.

\(p\propto{1\over V}\)

Pressure - at a surface and due to movement of particles

Pressure is the ratio of force acting to the surface area. The pressure of the hot gas is greater than the cold because the molecules hit the walls harder and more often due to their increased kinetic energy.

Standing waves

Standing waves (or stationary waves) form when the following occurs:

• Two waves
• With the same freqency and amplitude
• Travelling in opposite directions
• Interfere with each other

The most regular occurence of this is a single wave reflecting back onto itself.

Node: Point that has zero amplitude

Wavelength (λ): The distance between two equivalent points on consecutive waves (e.g. peak to peak)

Displacement-position graphs

\(v=f\lambda\)

Refraction

Critical angle: When light travels from glass to air, it refracts away from the normal. If the angle of incidence is large enough then the angle of refraction will be 90°. The angle at which this happens is called the critical angle.

NB: Sound not light. The angle of refraction can be calculated by Snell's law, where n is the refractive index (ratio of the speed of light to its speed in the material).

\({\sin i \over \sin r}={n_2 \over n_1}={v_1\over v_2}\)

Wavefronts

Electric circuits

Potential divider: Any circuit that splits the terminal potential difference of the battery using two or more resistors. 

Electromotive force or EMF (ε): The amount of work done per unit charge taking a small positive charge from one side of the battery to the other, when no current is being drawn.

Kirchhoff's second law: The sum of EMF around a closed loop = the sum of the potential difference

\(\sum \varepsilon = \sum V\)

This is a consequence of conservation of energy.

Resistance (R): The ratio of the potential difference across an electrical conductor to the current flowing through it

\(R={V\over I}\)

Resistance is measured in Ohms (Ω).

Volume

Volume will always be calculated by multiplying three lengths, sometimes with a numerical factor. This means it always has the unit m³.

Density

Density is defined as the ratio of the mass of a body to its volume:

\(\rho={m\over V}\)

Energy and power

Potential energy (E_p) is positional, equal to the amount of work done in taking a body from a place of zero potential energy to the place in question.

\(E_p=mgh\)

Power is defined as the rate at which work is done. It is a scalar quantity.

\(P={W\over t}\)

Atomic model

The Rutherford model of the atom had the following key ideas:

• A small, densely positive nucleus
• Surrounded by negative electrons
• Neutral overall

Nuclear physics

Beta particles consist of an electron. Electrons are not normally found in the nucleus, but are released during the process of beta decay, when when a neutron changes into a proton plus an electron. This reduces the ratio of neutrons to protons.

Binding energy is the amount of energy required to pull a nucleus apart into separate nucleons. It is equal to the energy released when the separate nucleons combine to form the overall nucleus. 

Beta decay example:

The \(\bar{\nu}\) symbol represents an anti-electron neutrino. The beta energy spectrum provides evidence of the neutrino, which takes some of the kinetic energy:

• Mass - almost nothing
• Charge - zero
• Spin - \(1\over 2\)