Radioactivity and Exponential Decay
- A model is a set of assumptions which simplifies and idealises a problem. The assumptions allow equations to be written, so calculations and predictions can be made
- Unstable atoms break down by releasing energy until they reach a stable form- this is radioactive decay and is a random process
- Radioactivity is modelled by exponential decay. An exponential decay curve is drawn through plotting the number of atoms decaying each second against time
- Activity (A)- the number of atoms decaying each second. It is proportional to size of sample; the activity falls as the sample size gets smaller. Is measured in becquerels (Bq)
- Decay constant (λ)- The gradient of a graph of the number of unstable atoms remaining against time is negative.
- Half-life (T1/2)- the average time for the number of undecayed atoms in an isotope to halve. Measured by measuring the time it takes for activity to half.
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- Capacitance (C) is the amount of charge stored per volt- a measure of how much charge a capacitor can hold. Capacitors store electrical charge.
- Capacitors are found in camera flashes, defibrillators and computer back up power supplies. They store charge for use when it is needed
- When a capacitor is set up in a circuit with a switch, when the switch is switched to the left charge builds up on the plates of the capacitor. The capacitor stores electrical energy.
- When the switched is flicked to the right, the plates discharge and charge is transferred to the bulb. Electrical energy is converted to light and heat
- Work is done when charge is removed from one plate and put onto the other. The energy for this is provided by the battery in the form of electrical energy
- The area under a graph of p.d. against charged stored on the capacitor gives the energy stored by the capacitor
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Charging and Discharging
- When connected to a battery, current flows to the capacitor until it is fully charged.
- Electrons flow onto the plate connected to the negative battery terminal => negative charge
- The build up of charge repels electrons off the plate connected to the positive battery terminal => plate positively charged.
- An equal but opposite charge builds on the plate, creating a potential difference. No charge flows between plates as they are separated by an insulator.
- Initially current is high, but falls as charge builds up. When pd across capacitor = pd across battery, current falls to zero and the capacitor is fully charged
- The battery can be removed to discharge the capacitor. When a charged capacitor is connected across a resistor, a current is produced which opposes the charging current.
- The time it takes to charge/discharge a capacitor depends on the capitance of the capacitor and the resistance of the circuit
- Discharge rate is proportional to charge remaining
- When discharging, charge on the capacitor falls exponentially- takes the same length of time to halve irrespective of the initial amount of charge
- Time constant (T) is time taken for the charge on a discharging capacitor (Q) to fall to 37% of Qo (charge on fully charged capacitor).
- Larger resistance = longer to charge/discharge capacitor
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- The natural log (ln) of undecayed nuclei against time gives a straight line (without log would be an exponential curve). The gradient of the line is -λ, which can be used to calculate half-life
- As many differential equations don't have exact solutions, iterative numerical methods are used to solve them.
- Iterative method for finding exponential discharge curve
- 1. Use charge to calculate discharge current
- 2. Use current x time to calculate ΔQ
- 3. Subtract ΔQ from Q to find the new charge
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Simple Harmonic Motion
- An oscillating system has repeated vibrations about a rest position, which have amplitude and frequency
- Simple Harmonic Motion (SHM)- an oscillation in which the acceleration of an object is directly proportional to its displacement from the midpoint, and is directed towards the midpoint
- As the object moves towards the midpoint, the restoring force does work on the object and transfers some PE to KE. When moving away from the midpoint, the KE is transferred back to PE
- At midpoint, KE = max and PE= 0. At maximum displacement, KE = 0 and PE= max
- Sum of KE and PE= mechanical energy. It is constant
- Cycle of oscillation= max positive displacement to max negative displacement
- IN SHM, frequency and period are independent of amplitude
- Frequency is directly proportional to time period
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Simple Harmonic Oscillators
- A mass on a spring is a simple harmonic oscillator. When the mass is pushed left or right of the equilibrium position, a force is exerted on it. F= kx
- When pushed/pulled away from equilibrium point, the spring is compressed or stretched so stores elastic potential energy. The area under a force-extension graph gives EPE
- The simple pendulum is another SHO
- The acceleration of an SHO is proportional to its displacement
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Free and Forced Vibrations
- All structures have a natural frequency which they will vibrate at. If the vibration is free, there will be no energy transfer to or from the surroundings, and the object will oscillate with the same amplitude forever. This does not happen in practice
- Forced vibrations occur when there is an external driving force. The frequency of the force is the driving frequency.
- Resonance is the vibration of an object with a rapidly increasing amplitude, and occurs when driving frequency= natural frequency
- Organ pipes, swings, glass smashing and radios all resonate
- Damping forces are frictional forces which cause an oscillating system to lose energy to the surroundings, and reduce the maximum amplitude. Systems can be deliberately damped to stop oscillations or minimise effects of resonance.
- The heavier the damping, the quicker the amplitude is reduced to zero. Critical damping reduces amplitude in the shortest possible time
- Lightly damped systems have a sharp resonance peak, and heavily damped systems have a flatter response- they aren't as sensitive to driving frequency and amplitude doesn't increase much
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