OCR B: Chapter 10

An atom is radioactive if it has

• Too many neutrons
• Not enough neutrons
• Too much energy

Radioactive decay is where unstable atoms break down by releasing energy and/or particles. It is a random process.

The model of radioactive decay is exponential decay and is based on a very large amount of atoms. 

The activity of a sample is the number of unstable atoms that decay each second. λ is the decay constant and is the probability of a nucleus decaying in a certain time.

The half life of an isotope is the average time it takes for the number of undecayedc atoms to halve. It can be measured from a graph.

To work out the activity of a sample: A = λN or (dN/dt = -λN)

 The solution of dN/dt = -λN is N=N0e-λt

To work out the half life you can use: T1/2 = ln2 / λ

You can work out the half life from the graph, as well as 2 times the half life etc.

Capacitors store charge to be used when it is needed. Capacitance is the amount of charge the capacitor can hold (C, measured in Farads). An example of a capcitor is a defibrillator.

The electrical energy that the capacitor stores is provided by the battery. A capacitor is charged due to a flow of electrons which come from the negative terminal of the battery. This means that a negative charge builds up on the plate connected to the negative, which repels electrons of the plate connected to the positive, making that plate positive. This means an equal and opposite charge builds up, meaning there is a p.d between the plates. The current in the circuit is initally high, but decreases since electrostatic repulsion makes it harder for electrons to be depositied. The capacitor is fully charged when its p.d = p.d across the battery. At this point the current in the circuit is zero.

A capacitor discharges when connected across a resistor. The p.d drives a current through the circuit which flows in the opposite direction to the charging current.  The capacitor is fully discharged when the p.d across the capacitor and the current in the circuit are zero.

The equations for capacitance is: C = Q/V

The equation for the energy stored is: E = 1/2QV = 1/2CV^2

The equation for the charge left is: Q=Q0e^-t/RC which is the solution to dQ/dt = -Q/RC

Another equation for capacitance is: C = εo.εr.A/d

SHM: an oscillation in which the acceleration of the object is directly proportional to the displacement from the micdpoint and directed towards the midpoint

Due to this restorative force, the objects will exchange PE and KE since as the object moves towards the midpoint, the restorative force does work so transfers PE to KE. The mechanical energy of a system is the sum of the PE and KE and will stay constant if the system is not damped.

The frequency of a system is the number of cycles per second, the period is the time taken for one complete cycle. In SHM the frequency and period are independent of the amplitude.

To find the velocity of a system, differentiate it's displacement. To find the acceleration of a system, differentiate it's velocity.

Simple harmonic motion occurs on pendulums and springs. 

If a system is oscillating and no energy is transfer to or from the system, it will keep oscillating with the same amplitude. Resonance happens when the driving frequency = natural frequency. As the driving frequency approaches the natural frequency, energy is transfered from the driving to driven system, increasing the amplitude.

Damping happens when energy is lost to the surroundings, which decreases the amplitude of oscillation. This can happen due to frictional forces or deliberately to minimise resonance.

Damping can be light, heavy, critical or over. A critcicaly damped system reduces the amplitude in the shortest possible time. An over damped system will take longer. 

Lightly damped systems have a sharp resonance peak since their amplitude only decreases rapidly close to the natural system. In heavily damped systems the ampltitude doesnt increase very much.

To find the frequency or period: f = 1/T

To find how the displacement varies with time: 

• x = Acos(2πft) (Starting at maximum displacement)
• x = Asin(2πft) (Starting at midpoint)

And remember x -> differentiate -> v -> differentiate -> a (Or intergrate in the other direction)

The force exerted on a spring: F = kx and therefore on a spring T = 2π√m/k

The elastic potenital energy of a spring is: E = 1/2kx^2

The formula for the period on a pendulum is: T = 2π√l/g

The differential equations for SHM are: d^2x / d^2t = -(2πf)^2 x and d^2x/dt^2 = -(k/m)x

You can draw logarithmic graphs of exponentials. For example, 

N =N0e^-λt

Taking logs: ln(N) = -λt + ln(N0)

And since y = mx+c you can work out various values from plotting this graph

A model is a set of assumptions that simplifies a problem and therefore makes it easier to make predictions and calculations.