EJC Physics H210 Oscillations 2023 1. Notes (2023) FULL For Microsite

Swings are one of a child’s earliest interactions with free oscillations and resonance. Find out why the frequency of oscillations does not differ with the mass of the person on it, and how our parents, siblings and friends naturally supply extra bursts of energy into the swing system under resonant conditions. Content • Simple harmonic motion • Energy in simple harmonic motion • Damped and forced oscillations, resonance Learning Outcomes Candidates should be able to: (a) describe simple examples of free oscillations (b) investigate the motion of an oscillator using experimental and graphical methods (c) show an understanding of and use the terms amplitude, period, frequency, angular frequency and phase difference and express the period in terms of both frequency and angular frequency (d) recall and use the equation 2ax =− as the defining equation of simple harmonic motion (e) recognise and use 0 sinx x t = as a solution to the equation 2ax =− (f) recognise and use the equations 0 cosv v t = and 22 0v x x=  − (g) describe, with graphical illustrations, the changes in displacement, velocity and acceleration during simple harmonic motion (h) describe the interchange between kinetic and potential energy during simple harmonic motion (i) describe practical examples of damped oscillations with particular reference to the effects of the degree of damping and to the importance of critical damping in applications such as a car suspension system (j) describe practical examples of forced oscillations and resonance (k) describe graphically how the amplitude of a forced oscillation changes with driving frequency near to the natural frequency of the system, and understand qualitatively the factors which determine the frequency response and sharpness of the resonance (l) show an appreciation that there are some circumstances in which resonance is useful, and other circumstances in which resonance should be avoided We have studied the kinematics of bodies in (i) straight line, (ii) projectile and (iii) circular motions. Oscillatory motions are another type of motion that can be often seen in everyday life: the swinging pendulum of a grandfather clock, the vibrations of a bus, and the waves on the strings of a guitar. Other non-visible examples include the electro-magnetic oscillations emitted by wireless equipment, gaseous atoms and molecules that propagate sound waves, and the lattice vibrations of atoms. To better understand how these systems work, it is important we learn the mechanics of oscillations.

If an oscillation repeats over time, it is a periodic motion. An oscillation begins when a system is perturbed from a condition of stable equilibrium. Force(s) tend to arise as a result of the initial disturbance, to bring the system back to the state of stable equilibrium – these are generally known as restoring forces. The initial perturbation introduces an amount of energy. Within the system, a continuous and periodic interchange of kinetic and potential energy takes place – there is usually maximal amounts of potential energy where/when the kinetic energy is at a minimal, and vice versa. Free oscillations occur when a system is displaced from its state of stable equilibrium and is then allowed to move or respond without restraint. There is no external force applied or resisting its motion. Therefore there is no gain or loss of total energy of the oscillating system. When a system oscillates freely, it does so at its natural frequency, which depends on some physical factors such as its dimensions, mass or elasticity. Free oscillations are oscillations with constant amplitude and without energy loss or gain as there is no driving or resistive forces acting on it. An oscillation is a to-and-fro motion between two limits Natural frequency is the frequency at which a body will vibrate when there is no driving or resistive forces acting on it. spring-mass stretched string pendulum (stable equilibrium) (stable equilibrium) (stable equilibrium) The natural frequency of a simple pendulum is dependent only on its length – so swings (look at it from the side) typically exhibit the same frequency regardless of the weight of the child.

We refer to a vertical spring-mass system attached to a ceiling to help familiarize ourselves with the various quantities used to describe an oscillation. equilibrium position position where no net force acts on the oscillating mass displacement x distance in a specified direction from equilibrium position of oscillating mass amplitude 0x maximum displacement from equilibrium position period T time taken for one complete oscillation of the oscillating mass frequency f number of complete oscillations per unit time. (unit: Hertz, Hz) 1f T = phase  an angular measure (in either degrees or radians) of the fraction of a cycle completed by the oscillating mass i.e. completion of one complete cycle corresponds to a phase 360 or 2 rad= phase difference  measure of how much an oscillation is out of step with another oscillation if two oscillations are in-phase: 0= out-of-phase: 0 anti-phase: OR 180 =  angular frequency  defined as the product of 2π and frequency. unit: radian s-1 (i) is effectively rate of change of phase of oscillating mass d dt  = (ii) shares same symbol  as circular motion’s angular velocity (rate of change of angular displacement d dt  =   but are different quantities! displacement x 0 time t + x0 - x0 period T period T equilibrium amplitude amplitude 2 f=

The oscillation of a vertical spring-mass system can be investigated using a motion sensor connected to a data logger. A motion sensor can measure position, velocity and acceleration of moving objects by emitting ultrasonic pulses and determining the time lag of pulses reflected back to the sensor. Because it needs a surface to reflect the ultrasound, it is not suitable if (i) the moving object is small (like a metal ball bearing) or (ii) soft (high absorbance of sound waves). • Set up the experiment as shown above. • Displace the mass vertically downwards and release it to set into vertical oscillations. • Measure the variation with time of height h using a motion sensor connected to a data-logger • Start the data-logger when the oscillations are steady. equilibrium position eqmh = 10.0 cm amplitude 0h = 2.0 cm period T = 3.2 s frequency f = 0.313 Hz angular frequency  = 1.96 rad s-1 displacement when t = 2.9 s ( )29h. = 1.0 cm equation for position h in cm* ( )ht = 2 1810 cos 362 21t. −  − *Start from ( )cos cos th = = . At t = 1.8 s, the function is a negative cosine of amplitude 2 spanning till t = 5 s, so ( ) 2cos 2 18h t.T =−  −   . Translate vertically up by 10 to yield equation. benchtop retort stand with boss and clamp spring oscillating mass motion sensor datalogger h time / s 5 15 0 1 2 3 4 5 h / cm 10

2ax =− Note: • 2 is a constant of proportionality between a and x • negative sign indicates that a and x are in the opposite directions SHM is mathematically the simplest case of free oscillations. Yet by itself, it allows us to gain a deep understanding of periodic motion and provides a basis to describe more complicated oscillations. SHM typically describes the motion of a single body. When multiple bodies (adjacent to each other in space) move individually with their own SHMs, we can witness progressive or stationary waves when we “zoom out” of the individual view and regard the many -bodies as a bulk system. A block of mass m rests on a smooth surface and is attached to the end of a light sprin