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Lesson 24 notes – Oscillation Graphs Objectives Be able to select and apply the equation vmax = (2πf)A for the maximum speed of a simple harmonic oscillator. Be able to describe, with graphical illustrations, the changes in displacement, velocity and acceleration during simple harmonic motion. Outcomes Be able to use the equation vmax = (2πf)A for the maximum speed of a simple harmonic oscillator correctly for different situations. Be able to rearrange and then use the equation vmax = (2πf)A for the maximum speed of a simple harmonic oscillator correctly for different situations. Be able to interpret graphical illustrations of the changes in displacement, velocity and acceleration during simple harmonic motion; Be able to derive the equation vmax = (2πf)A for the maximum speed of a simple harmonic oscillator Be able to draw graphical illustrations of the changes in displacement, velocity and acceleration during simple harmonic motion; Displacement – Time graphs Recap from lesson 6 and 5 Mechanics G481.1 on displacement time graphs and velocity time graphs. Imagine a mass on a spring being displaced and then dropping through the equilibrium position to as far down as it can go and back up again:Error! positive SHM Graphs If we let the mass hang it will stay in the equilibrium point. The strain of the spring equals the weight of the mass. If we pull the mass up, the strain is less and so it will accelerate towards the equilibrium point. If we pull it down the strain will be greater than the weight and so it will accelerate towards the centre again. At the equilibrium point the forces cancel leaving no acceleration and a maximum velocity. If we consider up positive then any displacement, velocity or acceleration directed downwards will be negative. The top graph shows displacement against time. The second graph shows the velocity of the oscillator against time and the third graph shows the acceleration against time. Maximum Velocity You can see that when the mass goes through the equilibrium point, x=0, and velocity is maximum. Maximum velocity of the oscillating system will happen when vmax = (2πf)A. When x is maximum, acceleration is negative maximum – the mass is being pushed back the opposite way (acceleration is directly proportional to negative displacement). This happens when the velocity is zero – since it is at its maximum amplitude. This isn’t easy so take some time going through it and drawing your own diagrams of an object oscillating about. The dotted green lines show how the graphs line up easily. You can see that the velocity graph is led by the displacement graph and is π/2 out of phase with it and the acceleration graph is led by the acceleration graph and is π/2 out of phase with that one. Explanation If we go back to the diagram we used to define some terms and look at the phasor diagram we can start to see clearly where the equations for SHM come from: Language to describe oscillations Sinusoidal oscillation +A Phasor picture s = A sin t amplitude A A angle t 0 time t –A periodic time T phase changes by 2 f turns per 2 radian second per turn = 2f radian per second Periodic time T, frequency f, angular frequency : f = 1/T unit of frequency Hz = 2f Equation of sinusoidal oscillation: s = A sin 2ft s = A sin t Phase difference /2 s = A sin 2ft s = 0 when t = 0 sand falling from a swinging pendulum leaves a trace of its motion on a moving track s = A cos 2ft s = A when t = 0 t=0 Motion of harmonic oscillator velocity force displacement against time against time against time large displacement to right right zero velocity mass m large force to left left small displacement to right right small velocity to left mass m small force to left left right large velocity to left mass m zero net force left small displacement to left right small velocity to left mass m left small force to right large displacement to left right zero velocity mass m large force to right left Dynamics of harmonic oscillator How the graph continues How the graph starts zero initial velocity would stay zero if no force velocity force changes velocity force of springs accelerates mass towards centre, but less and less as the mass nears the centre change of velocity decreases as force decreases new velocity = initial velocity + change of velocity trace curves inwards here because of inwards change of velocity t 0 0 trace straight here because no change of velocity no force at centre: no change of velocity time time Extension: More SHM Graphs You should learn to be able to understand and draw the following graphs: +2 r a a x +r -r t a = -r sin t -2r v x v = rcos t +r > 1 = 1 t v -r x t x = rsin t r is the amplitude in these graphs. And if you haven’t done it go back to the extension last lesson and try it. Use the diagrams below to help: Force, acceleration, velocity and displacement Phase differences Time traces varies with time like: displacement s /2 = 90 If this is how the displacement varies with time... cos 2ft ... the velocity is the rate of change of displacement... –sin 2ft ... the acceleration is the rate of change of velocity... –cos 2ft ...and the acceleration tracks the force exactly... –cos 2ft velocity v /2 = 90 acceleration = F/m same thing zero force F = –ks = 180 displacement s ... the force is exactly opposite to the displacement... cos 2ft Maximum Velocity Maximum velocity of the oscillating system will happen when vmax = (2πf)A. Let’s look at the equations. v= 2πfAcos(2πft) (a) and v=-A2πf sin(2πft) (b) are both solutions. Now, cos(2πft) or sin(2πft) have values of between 0 and 1, So to get the maximum for each of these we need cos(2πft) to be equal to 1 in (a) and sin(2πft) to be equal to 1 in (b). cos0 = 1 and sin(π/2) = 1 (if you’re not sure of this try it on your calculator) And if you look back at the graphs for displacement, velocity and acceleration you will see that velocity is a maximum when this occurs. If you can see the graphs in your head when you are tackling problems on SHM you will find them a whole lot easier.