Kinematics of Simple Harmonic Motion IB Physics 4.1

Why look at waves? Waves are the key to sound and colour. Mobile phone signals, microwave ovens all use energy carried by waves. Earthquakes and tsunamis are destructive waves of energy. Waves affect our everyday lives in many ways.

What is a wave? A wave is a pulse of energy. Waves carry energy away from a central transmitter. Mechanical waves, such as sound waves, need some medium of transmission. Electromagnetic waves, for example radio waves, can carry energy through a vacuum. If a wave is travelling through a medium, the particles of the medium do not move along with it. They vibrate about their equilibrium position, and the energy is transmitted through the interaction of neighbouring particles.

Transverse Waves

Water waves

Longitudinal Waves

Waves vs Oscillations A wave is travelling energy: all waves -- radio, light, x-ray, sound or water waves radiate in all directions from a central source. An oscillation is when a mass moves back and forth in a regular rhythm: a swing, the tide, a duck sitting still on a wavy pond all oscillate. Even though oscillations and waves are different phenomena, the same mathematical functions are used to describe them and graph their motion. These functions are the sine and cosine functions.

4.1.1Examples of Oscillating Systems A swinging pendulum, A mass in motion on a spring a vibrating string, a bobbing buoy or boat at anchor on a sea human vocal chords are all examples of oscillation

4.1.2 Wave Terms

v = f λ The Wave Equation v = velocity (ms-1) f = frequency (Hz) λ = wavelength (m)

Can you identify the parts of these waves?

Period and Phase Difference Wave 1 and 2 In Phase Period – this is the time it takes an oscillating system to make one oscillation. Phase difference - Wave 1 and 2 1800 out of Phase

Examples of SHM

Analysing SHM Movement

Analysing SHM

Springs and Simple Harmonic Motion X=0 X=A X=-A X=A; v=0; a=-amax X=0; v=-vmax; a=0 X=-A; v=0; a=amax X=0; v=vmax; a=0 X=A; v=0; a=-amax

Angular Frequency A useful quantity associated with SHM is angular frequency ω. This is defined in terms of the linear frequency as: ω = 2πf ω = 2π T Your turn: Change this to make T the subject!

Some other important SHM terms Amplitude (A): maximal displacement from the equilibrium. Period (T): the time it takes the system to complete an oscillation cycle. Inverse of frequency. Frequency (f ): the number of cycles the system performs per unit time (usually measured in hertz = 1/s). Angular frequency (ω): ω = 2πf Phase (θ): how much of a cycle the system completed (system that begins is in phase zero, system which completed half a cycle is in phase π). Initial conditions: the state of the system at t = 0, the beginning of oscillations.

Definition SHM Or a = -kx The negative sign indicates that the acceleration is towards the equilibrium position. If the acceleration a of a system is directly proportional to its displacement x from the equilibrium position, the system will execute SHM. We can express this definition mathematically as a = - constant x x

Tying in angular frequency ω Mathematical analysis using differentiation shows that the constant k is equal to ω2 where ω is the angular frequency of the system. Hence the equation now becomes a = - ω2x This equation is the mathematical definition of SHM.

Differential Equation of SHM Simple harmonic motion is also defined by the differential equation where "k" is a positive constant, "m" is the mass of the body, and "x" is its displacement from the mean position. This is a second- order differential equation. You do not need to know this. But it leads to ω = √k/m And therefore T = 2π√m/k

So the differential equation of SHM is… The second order differential of this general equation is d2x = - ω2x which also = a dt2 The solution of this equation is x = A sin (ωt) Where x is displacement from the mean position A = amplitude, ω = angular frequency in radians, t = time in seconds

What does moving along a circular path have to do with moving back & forth in a straight line (oscillation about equilibrium) ?? x = R cos q = R cos (wt) since q = w t y x -R R  1 2 3 4 5 6 x 8 8 q R 7 7

END – go to sheets

Helpful diagrams for sheets

x = A sin (ωt) This equation describes an object that moves back and forth along the x-axis with a maximum displacement A. 3 examples are shown Again using advanced calculus we can prove the solution described in the equation above. Differentiate once and we get dx = Aω cos (ωt) dt Differentiate again and we get d2x = - Aω2 sin (ωt) dt2 Since x = A sin (ωt) - as we saw earlier this can now be written as d2x = - ω2x which also = a

If a system is performing SHM then a Force F is required to bring about the acceleration a. The magnitude of the force is given as: F = -kx Again the negative sign indicates that the force is directed towards the equilibrium position of the system (In the case of masses causing the oscillation of springs this is often referred to as Hooke’s Law however the k above is different to the spring constant k of Hooke’s Law.)

Hooke’s Law In the case of a spring, k is the spring constant. The equation can be derived from Newton's second law (F = d(mv) / dt) as given by Hooke's law thus F=ma. If k is positive, the solution is a sinusoidal function (the exact form depends on the initial conditions ); It can be shown, by differentiation, exactly how the acceleration varies with time. Using the angular frequency ω, defined as ω = 2πf = 2π / T,