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SHM TAP Main aims • Recognize the characteristics of SHM. • State the condition required for SHM. • Use equations and graphs which represent the variation of displacement, velocity and acceleration with time. • Investigate mass-spring systems and the simple pendulum. • Discuss the effects of damping on SHM. • Describe energy changes during SHM. • State the conditions required for resonance to occur, and its effects. Consider … A pendulum. • Where is the equilibrium position? • Will the period of each oscillation change if the amplitude (starting point) changes? • How does the energy change as it oscillates? • If there was no friction, the amplitude (and total energy) would stay constant. Such oscillations (or near constant ones) are called free oscillations. • Does the period change as the pendulum slows down? • What force(s) is/are acting? Sketch the force at various parts of the oscillation. • Is it constant throughout the oscillation? • Consider a trolley between two springs or a mass on one spring. Would these be the same? • If we measure displacement from the equilibrium position, or midpoint, sketch the displacement – time graph for one complete oscillation from left to right and back. • Label +A, -A and T +A s = A sin t amplitude A A a 0 time t –A periodic time T phase changes by 2 f turns per second = 2f radian Periodic time T, frequency f, angular freque f = 1/T unit of frequency Hz = 2f Equation of sinusoidal oscillation: s = A sin 2ft s = A sin Phase dif s = A sin 2 s = 0 when sand falling from a swinging pendulum leaves a trace of its motion on a moving track s = A cos s = A whe t=0 Displacement X left +A Midpoint right -A Period T • What else does the graph tell us about the motion? • Is velocity constant throughout the oscillation? If not, how does it accelerate? • How can you tell that from the displacement-time graph? • Which way is it accelerating? • Draw a velocity time graph on the same axis (in a different colour) • When is velocity zero and maximum? • Draw an acceleration-time graph on the same axis. When is acceleration zero/max? Compare this graph with the displacement graph. • What would the force-time curve look like? When is force zero/max? Consider your graphs with reference to the other situations. How do they compare? • a mass bouncing on a spring • a trolley oscillating between two springs. A trolley between two springs SHM • All clocks (starting with the pendulum in a grandfather clock) are essentially simple harmonic oscillators • all transmitters and receivers of waves are essentially simple harmonic oscillators • many aspects of engineering design from the massive to the microscopic require a detailed knowledge of SHM (e.g. bridges, earthquake protection of buildings, atomic force microscopes for imaging single atoms) • Detailed theories of the behaviour of atoms and molecules (in solids and gases) are applications of SHM. Other aspects of SHM are at the heart of unsolved questions in physics today. Thus although the model oscillators that students meet appear rather basic, they mirror pretty well applications and problems far beyond the school laboratory Features of SHM • An object moving with SHM oscillates either side of a midpoint • The distance of the object from the midpoint = displacement • There is always a restoring force pulling or pushing the object back towards the midpoint • The size of the restoring force depends on the displacement and this force makes it accelerate towards the midpoint • the period is independent of the amplitude Displacement X left +A Midpoint right -A Period T Identifying SHM Using the apparatus of the earlier demonstration, set one oscillator in motion. Now change the amplitude. What do you notice about the time period of the oscillation? This is a characteristic of SHM; the period is independent of the amplitude, and we say that the motion is isochronic. Do these display SHM? 1. mass on a spring (vertical) 2. mass (large cube of polystyrene) on the end of a slinky spring suspended from the ceiling 3. mass between two springs (vertical, both springs in tension when the mass is at rest) 4. mass between two springs (horizontal, both springs in tension when the mass is at rest – use an air track slider for the mass to have a low friction system) 5. air track slider moving between two ‘buffer’ springs, or rebounding due to magnetic repulsionvibrating cantilever 6. simple pendulum 7. simple ‘half’ pendulum (one that bounces off a hard surface when its string is vertical) 1. simple pendulum whose string intercepts a peg when vertical, so the length of the pendulum gets shorter for one half of its cycle 2. a torsion pendulum 3. a ball bouncing off a hard surface 4. ball moving in a semi-circular shaped track (curtain rail) 5. ball moving in a vertical V shaped track (rounded enough at the point of the V to let the ball pass easily) 6. ball moving in a vertical parabolic shaped track (draw out the parabola on a large piece of paper to aid the bending of the track into the parabolic shape) 7. a right circular cone on an inclined plane 8. a rectangular or square section bar balanced on top of a cylinder – the length of the bar at right angles to the axis of the cylinder 9. water in a U tube 10. hydrometer in water • ‘simple’ pendulum (small mass on a string) • ‘compound’ pendulum (a rigid pendulum like a metre rule, with or without a mass on the end) • ‘torsion’ pendulum (a mass hanging from a single wire, executing twisting oscillations) • flexible strip of wood or plastic, clamped horizontally to a vertical support, oscillating in a horizontal plane • inertia balance (a version of the previous oscillator) • rolling ball on curved plastic tracking • mass oscillating on a vertical spring • large amplitude ‘pendulum’ (vertically rotating disc, pivoted at the centre with an off-centre mass clamped to it) • test tube ballasted and floating vertically in water • liquid in a large U-tube • vehicle on an airtrack between elastic barriers • ‘catapulting’ type oscillator (on an airtrack, or a trolley on a runway) Study some, or all, of the following oscillating systems: • ‘simple’ pendulum (small mass on a string) • ‘compound’ pendulum (a rigid pendulum such as a metre rule, with or without a mass on the end) • ‘torsion’ pendulum (a mass hanging from a single wire, executing twisting oscillations) • flexible strip of wood or plastic, clamped horizontally to a vertical support, oscillating in a horizontal plane • inertia balance (a version of the previous oscillator) • rolling ball on curved plastic tracking • mass oscillating on a vertical spring • large amplitude ‘pendulum’ (rotating disc, pivoted at the centre with an off-centre mass clamped to it) • test tube ballasted and floating vertically in water • liquid in a large U-tube • vehicle on an air track between elastic barriers • ‘catapulting’ type oscillator (on an air track, or a trolley on a runway). The oscillations will fall into three categories: • (a) SHM (i.e. constant period) (within the accuracy of observations) • (b) Approximately SHM, but departing at high amplitudes • (c) Definitely not SHM. It will probably emerge that (b) is the largest category. The main point of the activity is to demonstrate that many different oscillators behave in a way that is simple harmonic or nearly so. T T0 (1 2 16 ) • where T0 is the period for very small amplitudes and q is the angular amplitude in radians. For an amplitude of 1 radian (nearly 60°), T = 1.063T0, and for very lightly damped oscillations it ought to be possible to detect the 6% increase in period. Observations While you experiment on a particular oscillator consider the following questions: • Is the period constant (i.e. is it affected by the amplitude)? If the motion dies away quickly, you will need to think carefully how you will answer this point. • Can you identify separate stiffness (k) and inertia (m) components? If you cannot easily do this, can you pick out features of the system that behave like k and m (and try and say in a sentence or two why you found it difficult)? • What adjustments do you think you would need to make to the system to reduce its period (increase its frequency)? Motion of harmonic oscillator displacement 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 velocity against time force against time 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 Force, acceleration, velocity and displacement Phase differences Time traces varies with time like: displacement s /2 = 90 /2 = 90 = 180 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 acceleration = F/m same thing zero If this is how the displacement varies with time... force F = –ks displacement s ... the force is exactly opposite to the displacement... cos 2ft Dynamics of harmonic oscillator How the graph starts zero initial velocity would stay zero if no force velocity force changes velocity How the graph continues 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 time trace straight here because no change of velocity no force at centre: no change of velocity time Uniform Circular Motion (radius A, angular velocity ) Simple Harmonic Motion (amplitude A, angular frequency ) Simple harmonic motion can be visualized as the projection of uniform circular motion onto one axis. The phase angle t in SHM corresponds to the real angle t through which the ball has moved in circular motion. • Angular frequency ω = 2π/T = 2πf • Phase difference = 2πΔt/T a = - (2πf)2x. The proof for the constant of proportionality (2πf)2 is on p38. 1/8 2π or 2πt/T or 2πft When object is half way displacement X = Acos(2π/2) Or Acos (π) = -1A Displacement x (from the equilibrium point) after time t = Acos(2πft) when the motion started at Xmax = A, with v=0. 1/8 2π or 2πt/T When object is at mid point displacement X = Acos(2π/4) Or Acos (π/2) = 0 X = Acos(2π/8) Using this formula to calculate displacement you must be in radian mode!