What do these two have in common? They both move back and forth about a central point and are examples of PERIODIC MOTION

Describing Motion Which words would you use to describe these motions? Regular movements like this are called OSCILLATIONS. A pendulum moves back and forth with a regular beat, even as the oscillations die away, it’s time period (time for one complete oscillation) stays the same.

Time Period, T Time Period: This is the time taken for one complete oscillation/cycle. Equilibrium position Positive displacement Negative displacement One complete oscillation is classed as that when the pendulum returns to it’s original position

Oscillations History: In 1581, Galileo watched lanterns in a cathedral swing back and forth. He noticed the time for each swing stayed the same even as the swinging died down. Galileo realised that this phenomena could be used as a means of time keeping. Oscillating pendulums still rely on this method where as modern quartz watches rely on vibration of quartz crystals.

Oscillations A pendulum moves back and forth with a regular beat, even as the oscillations die away, it’s time period (time for one complete oscillation) stays the same. Why? As the amplitude of the swing decreases, the pendulum moves a smaller distance at a slower speed and the time, T stays the same.

SHM When an object oscillates with constant time period even if the amplitude varies, we say it is moving with Simple Harmonic Motion (SHM).

SHM in a swing How does the velocity and the acceleration of the swing change as it swings backwards and forwards? At the end of each oscillation the swing is momentarily stationary. It then steadily speeds up as it approaches the centre before beginning to slow down again. As velocity is a vector we take the forward direction as positive and the backwards direction as negative.

SHM When at the highest point in the swing you speed up towards the centre. Once past the centre you begin to slow down until you stop at the highest position. Deceleration is the same as negative acceleration so you are still accelerating towards the centre.

SHM For an object moving with SHM: 1)The acceleration is always directed towards the equilibrium position at the centre of the motion. 2)The acceleration is directly proportional to the distance from the equilibrium position.

Definition of SHM The motion where the acceleration (or force) is directly proportional to the displacement from a fixed point and is always directed towards that fixed point.

Displacement-time graph The displacement-time graph for SHM is sinusoidal. It is the shape of a sine or cosine curve. One oscillation is the time period, T Amplitude, A, this is the maximum displacement and is measured from the centre of the oscillation A - A 0 Displacement f = 1 / T T = 1 / f rd.ac.uk/feschools/waves/ shm.htm

Velocity-Time graph You can deduce a velocity-time graph for SHM from the gradient of the displacement-time graph, (because velocity is the rate of change of displacement). The gradient of the D-T graph is zero when the displacement is a maximum. The velocity is zero at these points. The gradient of the D-T graph is a maximum when the displacement is zero. This is when the object is in the centre of its motion. The velocity is a maximum at this point (positive or negative).

Acceleration-Time graph You can deduce an acceleration-time graph for SHM from the gradient of the velocity-time graph, (because acceleration is the rate of change of velocity). The gradient of the V-T graph is zero when the velocity is a maximum. The acceleration is zero at these points (the middle of oscillation) The gradient of the V-T graph is a maximum when the velocity is zero. This is when the object is at its highest point. The acceleration is a maximum at this point.

Relationship between acceleration and displacement Look at the graphs you have drawn of displacement and acceleration. What do you notice? acceleration = 0 when displacement = 0 (in the middle of the motion) acceleration is large when displacement is large (but to left when displacement is to the right) acceleration is max when displacement is max (but to left when displacement is to the right)

Relationship between acceleration and displacement The acceleration is proportional to the displacement, but in the opposite direction to the displacement. Acceleration α – displacement or a α - x

Relationship between acceleration and displacement We know: a α – x We can rewrite this equation by inserting a constant of Proportionality. a = - (2πf) 2 x or a = - ω 2 x Where f = frequency of oscillation N.B. ω 2 is always positive, so the acceleration and displacement are always of opposite signs.

Linking displacement x and time t As we know the displacement – time graph for SHM is sinusoidal. Therefore the equations for this shape of graph must be sine or cosine. In fact: If timing starts at the CENTRE of the oscillation: X = A sin (2π f t) If timing starts at the maximum displacement: X = A Cos (2π f t) Where A is amplitude of oscillation and f is frequency. N.B. Angles are in RADIANS

Linking displacement x and velocity v If an object is moving with SHM then its velocity at any point can be found using: V = +/- 2π f √(A 2 – x 2 ) We can use this equation to find the maximum velocity during SHM, i.e. when displacement x = 0 – at the equilibrium position. V max = +/- 2π f A

The Simple Pendulum A pendulum oscillates with SHM provided the amplitude is small. The time period of a pendulum depends on one thing – the length of the string. As the length of the string increases the time period increases. The time period of a pendulum is found from: T = 2π√ ( l /g) (Where l is the length of the pendulum, measured to the centre of the bob.) We can use a simple pendulum to measure ‘g’ accurately. We can measure the time period T using a stopwatch, measure the length of the pendulum l in metres and rearrange the equation to calculate ‘g’. Alternatively we could plot T against √l giving a straight line through the origin. The gradient of this line is equal to 2π/√g. (Prac 4.1).

A Mass on a spring To find the time period T for a mass m oscillating on a spring we use: T = 2π √(m/k) Where k is the spring constant - the stiffness of the spring. This is constant for a particular spring and is measured in Newtons per metre (N/m).

A Mass on a spring Connect a mass to a spring, pull it down and let go. What happens? The mass will bounce up and down about its equilibrium position – oscillates with SHM. What affects the time period of this oscillation? 1)The mass – the greater the mass the slower it accelerates therefore the greater the time period. 2)The stiffness of the spring – a stiffer spring pulls the mass back to its equilibrium position with a greater force. Therefore its moving faster and the time period decreases.

The stretching force on a spring (Hooke’s Law) If the stretching force on a spring increases by equal amounts then the spring will extend by equal amounts. This is called Hooke’s Law. Stretching Force (F) = Spring Constant (K) x Extension (x) Provided the elastic limit is not exceeded, the elastic potential energy stored in the spring can be calculated using: Elastic Potential Energy = ½ Stretching Force x Extension (the area under a Force : Extension graph) By substituting the first equation into the second, we get. Elastic Potential Energy = ½ Spring Constant x (Extension) 2 Ep = ½ KX 2

Phase Angles Imagine two masses on springs bouncing up and down side by side. How can these oscillations differ? They could have different periods and amplitudes. But even if these are the same, one could be at it’s max +ve displacement and one could be at it’s max –ve displacement. They could be out of step with one another. We use the term PHASE to describe how out of step two oscillations are. One complete oscillation has a phase of 2π radians. If the two objects are at exactly the same point at all times we say that they are in phase and that the phase difference is zero. If the two objects are oscillating ½ cycle out of step, we say that they are in anti-phase and that the phase difference is π rad. When is the phase difference π/2 rad?

Energy in SHM Think of yourself on a swing. At which points will you have maximum and minimum kinetic energy? At the maximum amplitude of your swing you will be stationary for a moment therefore KE is zero. You will be at your fastest as you pass the centre (equilibrium position) and at this point KE will be a maximum. If energy is always conserved, where does this increase in KE come from? At maximum amplitude your gravitational potential energy is a maximum and as you pass the centre the minimum. All SHM involves the continual transfer between kinetic and potential energy. The total amount of energy remains constant.

Energy and Amplitude FACT: The more we pull down a mass on a spring the bigger the oscillations. QUESTION: If we double the amplitude of the oscillations does the energy double? Lets think about the energy in the system! As the mass moves through it’s equilibrium position it has maximum kinetic energy. KE = ½ mv 2 and V max = +/- 2π f A Therefore: KE = ½ m (2π f A) 2 and so KE α A 2 If we double the amplitude of the oscillations the energy increase by 4 times.

Mechanical Resonance If you want to make a swing go higher, how do you time your pushes? You push in time with the swing’s movement. You match the frequency of your pushing force with the natural or resonant frequency of the swing. We can force objects to vibrate at any frequency but all oscillating systems have their own natural frequency. If the driving frequency is the same as the natural frequency then the amplitude builds up. This effect is called resonance. Resonance can be destructive or useful ! DestructiveUseful Tacoma Narrow Bridge disaster Tuning circuits in TV/radio Opera singers shattering wine glasses Wind instruments

Damping So far we have assumed that no energy is lost from an oscillating system and that is continues to oscillate indefinitely. We know that in practice the system begins to slow down and eventually stops. Why? because air resistance slows it down. Energy is lost from the system in overcoming this friction. This effect is called damping. If the oscillations take a long time to die away we call it light damping. If the oscillations die away very quickly, e.g. swinging a pendulum in water we say it is heavily damped. Damping that allows an object to move back to it’s equilibrium position as quickly as possible is called critical damping. An example of this is ‘shock absorbers’ in car suspension. If the system is over-damped then it does not oscillate.

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