An oscillation is a simple repetitive to and fro movement. A child’s spring just like a string on a guitar is an example of free vibration. Once displaced from its position, it vibrates at its own natural frequency.
· In a free vibration there is a constant interchange between potential (stored) energy and kinetic energy.
· A swing reaches zero kinetic energy an maximum gravitational potential energy when at its greatest displacement, the kinetic energy is at a maximum and the gravitational potential energy at a minimum when the displacement equals zero.
After a swing is given a push, it vibrates at its natural frequency. To make it swing higher subsequent pushes need to coincide with the vibrations of the swing, This is an example of resonance, the large amplitude oscillation that occurs when the frequency of a forced vibration is equal to the natural frequency of the vibrating object.
Resonance happens when the driving force = the natural frequency. As the driving force approaches the natural frequency, the system gains more and more energy from the driving force and so vibrates with rapidly increasing amplitude. When this happens the system is resonating.
Just like linear motion, all mechanical oscillations are subjected to resistant forces. Damping is the loss of energy to the environment. The effect of resistive force in removing energy from a vibrating object is known as damping. A string vibrating in air is lightly dampers as the main resistive force is air resistance. The effect of light damping is to gradually reduce the energy, and therefore the amplitude, of the vibrating object.
The body of a car is connect to the wheels by springs. When the car goes over a bump the spring compress, giving the rider a smoother ride. If the car was lightly damped the car would continue oscillate after going over the bump. Shock absorbers increase the resistive force, so that when the body of the car is displaced it returns to it original position without oscillating. A car is said to be critically damped so it returns to its original position without oscillation in the minimal amount of time.
Very large resistive forces result in heavy damping. Imagine a mass on a spring suspended so that the mass is in a viscous liquid such as syrup. If the mass is displaced, the force opposing its movement is very large and it takes a long time to return to the normal position.
Simple Harmonic Motion:
As the name implies it is the simplest kind of oscilatioty motion. One set of equations can be used to describe and predict the movement of any obhect whose motion us simple harmonic:
· Its acceleration is proportional to its displacement.
· Its acceleration and displacement are in opposite directions.
The second bullet point means that the acceleration, and therefore the resultant force, always acts towards the equilibrium position, where the displacement is zero. Also the size of the resultant force depends on the displacement and the force makes the object accelerate towards the midpoint.
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.
The restoring force makes the object exchange potential energy and kinetic energy:
1. The type of potential energy depends on what it is that’s providing the restoring force. This will be gravitational potential energy for a pendulum and elastic potential energy for masses on a spring.
2. As the object moves towards the midpoint, the restoring force does work on the object and so transfers some potential energy to kinetic energy. When the object is moving away from the midpoint, all that kinetic energy is transferred back to potential energy.
3. At the midpoint, the object’s potential energy is zero and its kinetic energy is maximum.
4. At the maximum displacement (the amplitude) on both sides of the midpoint, the objects kinetic energy is zero and it potential energy is maximum.
The frequency and period don’t depend on the amplitude:
1. From the maximum positive displacement (eg maximum displacement to the right) to maximum negative displacement (eg. Maximum displacement to the left) and back again is called a cycle of oscillation.
2. The frequency, f, of SHM is the number of cycles per second (measured in Hz).
3. The period, T, is the time take for a complete cycle (in seconds),
4. The relationship between frequency and period is given by the equation:
F = 1/ T
In SHM, the frequency and period are independent from the amplitude (ie. Constant for a given oscillation). So a pendulum clock will keep ticking in regular time intervals even when the swing becomes very small.
SHM Equations 1:
1. According to the definition of SHM, the acceleration, Δ²x/Δ²t, (also –(2∏f²) is directly proportional to the displacement x. The constant of proportionality depends on the frequency, the acceleration is always in the opposite direction from the displacement ( hense the minus sign)
2. The velocity is positive when the object is moving in one direction, and negative when it’s moving in the opposite direction. For example a pendulum’s velocity is positive when it’s moving from left to right and negative when moving from right to left.
3. The displacement varies with time according to two equations, depending on where the object was when the timing was started.
For someone who has started a stopwatch wit a pendulum at maxium displacement: x=Acos(2∏ft)
For someone releasing a pendulum but starting a stopwatch as the pendulum swings through the midpoint:
x = Asin(2∏ft)
SHM Equations 2:
Simple Harmonic Oscillation:
1. When the mass is pushed to the right of the equilibrium position, there’s a force exerted on it. The size of the force is : F = kx. Where k is the spring constant
2. After a bit of hookahs pookas involving Newton’s second law, F=ma and other **** you get the formula for the period of a mass oscillating on a spring.
T = 2∏Öm/k
The activity, or rate of decay, of a sample of radioactive material is measured in Becquerel (Bq). An activity of 1 Bq represents a rate of decay pf 1 s-1.
Radioactive decay is a random process and the decay of an individual nucleus cannot be predicted. However, given a sample containing large numbers of undecayed nuclei, then statistically the rate of decay should be proportional to the number of decayed nuclei present. Double the size of the sample and, on the average, the rate of decay should also be doubled.
There are two factors that effect the rate of decay of a sample of radioactive material: They are:
· The radioactive isotope involved
· The number of undecayed nuclei.
The relationship between the rate of decay, or activity, of a radioactive isotope and the number of undisclosed nuclei is:
Activity, A = λN
When the activity is measured in Becquerel, N is the number of undecayed nuclei present and λ is the decay constant of the substance, λ has units s-1.
An alternative way of expressing activity is the rate of change undecayed nuclei with time:
ΔN/Δt = -λN
This leads to the relationship between the number of undecayed nuclei, N, and the number at time t=0, N0:
The half life of a radioactive isotope ½t is the average time taken for the number of nuclei of the isotope to halve.
The half life can be calculated using the equation:
There is a relationship between the half-life ad the decay constant of any particular radioactive isotope. The shorter the half life the greater the decay constant and greater the rate of decay.