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PowerPoint used in Lesson

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11 MODELLING OSCILLATIONS LEARNING OBJECTIVES

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AQA Physics A-level Section 6.1: Further Mechanics Revision - PMT

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Notes, flashcards, videos and past exam questions by topic for AQA Physics A-Level Section 6.1 - Further Mechanics

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Simple Harmonic Motion arises when we consider the motion of a particle whose acceleration points towards a fixed point O and is proportional to the distance of the particle from O (so the acceleration increases as the distance from the fixed point increases).

As the particle moves away from the fixed point O, since the acceleration is pointing towards O, the particle will slow down and eventually stop (at Q), before returning to O. It will keep going and then again slow down as it reaches P before stopping at P and returning to O once more.

The particle will therefore move between two fixed points (P and Q). The amplitude of the motion is the distance from O to either P or Q (the distances are the same).

A particle which moves under simple harmonic motion will have the equation

where w is a constant (note that this just says that the acceleration of the particle is proportional to the distance from O).

You may be asked to prove that a particle moves with simple harmonic motion. If so, you simply must show that the particle satisfies the above equation.

We can solve this differential equation to deduce that:

where v is the velocity of the particle, a is the amplitude and x is the distance from O.

From this equation, we can see that the velocity is maximised when x = 0, since v ² = w ²a ² - w ²x ²

Hence the maximum velocity is aw (put x = 0 in the above equation and take the square root).

The period of the motion is the time it takes for the particle to perform one complete cycle. It can be calculated using:

If the particle is at 0 when t = 0, then the following equation also holds:

If the particle is at P or Q when t = 0, then the following equation also holds:

A simple pendulum consists of a particle P of mass m, suspended from a fixed point by a light inextensible string of length a, as shown here:

So we have approximate simple harmonic motion, where w ² = g/l .

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Simple Harmonic Motion | a2-level-level-revision, maths, mechanics, simple-harmonic-motion | Revision World

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Simple Harmonic Motion arises when we consider the motion of a particle whose acceleration points towards a fixed point O and is proportional to the distance of the particle from O (so the acceleration increases as the distance from the fixed point increases).

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Simple Harmonic Motion

You may have heard the one about the opera singer breaking a wineglass with their voice. (In other versions it's a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations. Later, we'll show you how to break a wineglass with sound, but first we'll try to work out some rules for things which vibrate, or oscillate.

There are many forms of oscillation in the real world - oscillations determine the sound of a musical instrument, the colour of a rainbow, the ticking of a clock and even the temperature of a cup of tea. A mechanical oscillation is a repeating movement - an electrical oscillation is a repeating change in voltage and current.

Mechanical oscillations are probably easiest to think about, so we'll start there. Let's think about the movement of an object back and forth over a fixed range of positions, such as the movement of a swing in a playground, or the bouncing up and down of a weight on the end of a spring (this last one is one of those examples which teachers / scientists / engineers love, but which doesn't seem to have anything to do with the real world. As we'll see, it does!!).

The pendulum and the mass-spring system are both oscillating. How can we describe their oscillation?

Displacement

The animation shows how displacement describes the distance (and also direction as displacement can be negative as well as positive) of the object from its equilibrium position.

The displacement amplitude tells us how 'big' the oscillations are. We can use the peak value (the maximum positive displacement from the equilibrium position) or the peak-to-peak value (the distance between biggest negative and biggest positive displacement).

The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.

Can you see why the first time the mass returns to the equilibrium position, we've only reached half a cycle?

Displacement on its own will sometimes do. But often it is also useful to understand the oscillation in terms of velocity or acceleration. These three things are very similar.

When our oscillating object reaches maximum displacement (when it is as far from equilibrium as it can get) it changes direction. This must mean that there is an instant in time when it is not moving. So, at the point of maximum displacement, its speed is zero. Then it speeds up in the opposite direction, and travels fast through the equilibrium position before starting to slow again in preparation for the next change in direction.

Velocity is just another word for speed, with the extra feature that it has direction and therefore can be negative or positive. (If you walk backwards at 5 km/h. - difficult without falling over - then your velocity is minus 5 km/h.)

So - when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity are shifted compared to the other. This shift along in time is called a phase angle, which is an idea we'll come back to later.

At maximum displacement the object has to come to rest, and then start moving in the opposite direction. Imagine braking in a car (moving forwards) before immediately moving off fast in reverse - you'd feel rapid deceleration during braking, followed acceleration in reverse gear. Like displacement and velocity, acceleration can be negative or positive depending on direction. In this example, braking would cause negative acceleration, as would moving off in reverse. It sounds weird, but the point in time where the acceleration would be (negative) maximum is the moment when the car stops moving - just as it changes direction.

Thinking about our animated mass-spring system, we can see this effect. Conversely as the object moves through its equilibrium position, the velocity is only changing very slowly and the acceleration actually becomes zero. This is shown in the acceleration vs time graph below:

Note again - the period is the same, but things don't line up in time. There's some phase changein there too.

In our animations, the mass on a spring bounces up and down with a regular pattern or waveform. Why does this happen? Let's think about playground swings. What's the first thing we do to get the swing moving?

The swing is first pulled back, and then pushed forward. It then oscillates back and forth 'on its own' until it slowly comes to rest at its equilibrium position - the middle position between the two extremes of displacement. This is usually the place where an object will naturally rest if no external forces are applied to it.

Once the swing has been pulled away from its equilibrium position, the force of gravity will act to bring it back. This force will always act in a direction towards the equilibrium position, and is known as a restoring force. If the swing is pulled higher into the air (larger displacement amplitude) then there will be a larger restoring force acting on it, and when the pusher lets go it will travel further. This shows that the restoring force is proportional to the distance of the swing from the equilibrium position.
Towards the end of the the animation the forces are shown.

So why doesn't the swing just return directly to the equilibrium position, and stay there?

The restoring forces are large enough to make the swing overshoot. This is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to overshoot equilibrium and travel up the other side. The process keeps repeating and an oscillation is formed. If no other forces are applied to the system, the swing will keep swinging back and forth forever, with a constant period and constant amplitude.

Does this happen in real life? (Silly question - of course not). Then why not?

In all real-life oscillating systems, damping comes into play. Damping is all about the loss of energy, often due to resistance or friction. The air displaced by the swinger adds some resistance, and even if the swing were in a vacuum (and the swinger were, therefore, dead) the pivots at the end of the chain generate some friction with their supports and slowly, almost immeasurably, heat up. This uses energy, which has to come from somewhere. And so the amplitude of oscillation decreases.

If you have no pusher, and have to propel yourself on the swing, you have to put some effort into it to keep the swing moving. This effort is overcoming damping. If there were no damping (and also if you could switch off your heart, brain and other energy-consumptive body functions) you could swing forever without needing to eat!
Next: Inertia

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You may have heard the one about the opera singer breaking a wineglass with their voice. (In other versions it's a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations.

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This applet plots the motion of a driven simple harmonic oscillator. The user can control the frequency of the vibrator, the spring constant, the degree of damping, and the mass.

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11 MODELLING OSCILLATIONS LEARNING OBJECTIVES

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Animation shows two blocks of equal mass connected to rigid supports by stiff springs (white). The two masses are also connected by a weak spring (red) to each other. Initially the right mass is drawn to the right holding the left mass at its equilibrium position.

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Kinematics Graphs for Simple Harmonic Motion

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The dog dies!

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Video of a resonance test of a military helicopter. I am glad that they test for these kinds of things while still on the ground!

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Slow-motion video of a glass that breaks because of resonance. The sound waves (which you can not hear in this video) match the natural frequency of the glass

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Two masses attached to a spring are oscillating on a smooth horizontal surface. The spring is initially in an elongated state and pressing start button releases the masses. Graphs of position vs time of the two masses are shown. Notice that when the masses are unequal the frequencies of oscillations are same but the amplitudes differ.

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Have some fun. You may like to try this. Set view to horizontal plane, set yellow ball's mass and velocity to maximum and red ball's velocity to minimum. Also try the other extreme of setting yellow ball's mass to minimum and red ball's velocity to maximum.

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Forced Damped Oscillations Simulation

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Oscillating body is initially at the top extreme position. The amplitude and the frequency of oscillations can be adjusted using the sliders. The black vector is the force on the body due to the spring. A periodic force whose frequency and amplitude can be varied can be applied on the body.

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The purpose of the animation is to highlight impotance of phase difference when oscillations are added. Two SHM's along the x-axis of differing amplitudes and frequencies are being added.Motion of the green particle and yellow particles are The red particle (when show sum check box is ticked) shows the resultant motion.

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Mass-Spring Oscillations Animation

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The animation shows oscillations of a mass attached to a spring. It is very instructive to study the graphs when the damping coefficient is not equal to zero.

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Phase Animation

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This animation shows number of particles in SHM. All the particles have same amplitude, same frequency but are at different positions at any given moment. The term which would deal with this difference effectively is phase.

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The red particle is going round the circular path in anti-clock wise sense (considered positive) with a uniform speed and its x-projection and y - projection are shown in cyan and orange respectively.

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The Spring Pendulum model displays the model of a hollow mass that moves along a rigid rod that is also connected to a spring.  The mass, therefore, undergoes a combination of spring and pendulum oscillations.

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The applets shows a mass connected to a horizontal spring on a surface with friction. The user can control the  degree of damping.

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The damping factor may be controlled with a slider. The maximum available damping factor of 100 corresponds to critical damping.

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The Pendulm Energy Model shows a pendulum and associated energy bar charts. Users can change the initial starting point of the pendulum

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The Simple Pendulum model displays the dynamics of a simple pendulum.  The pendulum is initially displaced from equilibrium and the pendulum bob has zero initial velocity.

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Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It's easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of g on planet X. Notice the anharmonic behavior at large amplitude.

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This applets plots the displacement, velocity, and acceleration of a mass connected to a vertical spring. This applet is excellent at showing the relationship between SHM and circular motion

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A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.

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Illustrating and comparing Simple Harmonic Motion for a spring-mass system and for a oscillating hollow cylinder.

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Demonstrating that one component of uniform circular motion is simple harmonic motion.

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Resonance of a damped driven harmonic oscillator

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The largest response to the drive force occurs when the natural solutions are underdamped and the drive frequency is the same as the natural frequency $\omega_0$. This condition is called a resonance. For lightly damped systems, the drive frequency has to be very close to the natural frequency and the amplitude of the oscillations can be very large.

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Phase Graph Simulation

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Oscillations and Simple Harmonic Motion. Mechanics with animations and video film clips. Physclips provides multimedia education in introductory physics (mechanics) at different levels. Modules may be used by teachers, while students may use the whole package for self instruction or for reference.

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The Coupled Oscillators and Normal Modes model displays the motion of coupled oscillators, two masses connected by three springs.

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A harmonic oscillator driven by a harmonic force. The frequency and damping factor of the oscillator may be varied

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Damped Driven Simple Harmonic Oscillator

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Damped Driven Simple Harmonic Oscillator

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The Damped Driven Simple Harmonic Oscillator model displays the dynamics of a ball attached to an ideal spring with a damping force and a sinusoidal driving force.  The spring is initially unstretched and the ball has zero initial velocity.

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SHM for a bouncing mass from first principles