OCR A Physics A-level Topic 5.4: Oscillations Notes PDF

OCR A Physics A-level Topic 5.4: Oscillations Notes www.pmt.education  Simple harmonic motion Key definitions Displacement, x – the distance from the equilibrium position Amplitude, A – the maximum displacement Period, T – the time t...

OCR A Physics A-level Topic 5.4: Oscillations Notes www.pmt.education  Simple harmonic motion Key definitions Displacement, x – the distance from the equilibrium position Amplitude, A – the maximum displacement Period, T – the time taken to complete one full oscillation Frequency, f – the number of oscillations per unit time Phase difference, ϕ – the fraction of an oscillation between the position of two Δ𝑡𝑡 oscillating objects (given by 𝑇𝑇 × 2𝜋𝜋) Angular frequency, ω – the rate of change of angular position (given by 2𝜋𝜋𝜋𝜋) Simple harmonic motion Simple harmonic motion is a type of oscillation, where the acceleration of the oscillator is directly proportional to the displacement from the equilibrium position, and acts towards the equilibrium position. The key equation for simple harmonic motion is 𝑎𝑎 = − 𝜔𝜔2 𝑥𝑥 where a is the acceleration of the oscillator, ω is the angular frequency, and x is the displacement of the oscillator. The negative sign is used to show that the direction of acceleration is always towards the equilibrium position, in the opposite direction to the displacement. An oscillator in simple harmonic motion is an isochronous oscillation, so the period of the oscillation is independent of the amplitude. Techniques to investigate the period and frequency of simple harmonic motion The frequency of the oscillator is equal to the reciprocal of the period. The period of the oscillator, and hence the frequency, can be determined by setting the oscillator (such as a pendulum or a mass on a spring) in to motion, and using a stopwatch to measure the time taken for one oscillation. In order to increase the accuracy of this measurement, the time for 10 oscillations to take place can be measured, and this time divided by 10 to find the period. An oscillator in simple harmonic motion is an isochronous oscillation, so the period of the oscillation is independent of the amplitude. A fiducial marker is used as the point to start and stop timings, and is normally placed at the equilibrium position. Analysing simple harmonic motion There are two equations which can be used to determine the displacement of a simple harmonic oscillator. 𝑥𝑥 = 𝐴𝐴 sin 𝜔𝜔𝜔𝜔 𝑥𝑥 = 𝐴𝐴 cos 𝜔𝜔𝜔𝜔 www.pmt.education where x is the displacement of the oscillator, A is the amplitude, 𝜔𝜔 is the angular frequency, and t is the time. The sine version of the equation is used if the oscillator begins at the equilibrium position, and the cosine version is used if the oscillator begins at the amplitude position. Velocity and acceleration The velocity of the oscillator at a given time can be determined by finding the gradient of the graph at that point. The maximum velocity occurs at the equilibrium position, with the oscillator being stationary at the amplitude points. The maximum acceleration occurs at the amplitude points, and is 0 when the oscillator is at equilibrium position. The velocity, v, of the oscillator is given using the equation 𝑣𝑣 = ±𝜔𝜔 𝐴𝐴2 − 𝑥𝑥 2 where 𝜔𝜔 is the angular frequency of the oscillator, A is the amplitude, and x is the current displacement. The maximum velocity occurs at the equilibrium position, where x = 0, so we can derive the formula 𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜔𝜔𝜔𝜔 to determine the maximum velocity of the oscillator. Energy transfers in simple harmonic motion Interchange between kinetic and potential energy During simple harmonic motion, energy is exchanged between the kinetic and potential forms. The maximum kinetic energy occurs at the equilibrium point, where the velocity is at a maximum. The maximum potential energy occurs at the amplitude positions, where displacement is at a maximum. Total energy is always conserved. Damping Damping is the process by which the amplitude of the oscillations decreases over time. This is due to energy loss to resistive forces such as drag or friction. Light damping occurs naturally (e.g. pendulum oscillating in air), and the amplitude decreases exponentially. When heavy damping occurs (e.g. pendulum oscillating in water) the amplitude decreases dramatically. In critical damping (e.g. pendulum oscillating in treacle) the object stops before one oscillation is completed. www.pmt.education  Resonance Free and forced oscillations When an object oscillates without any external forces being applied, it oscillates at its natural frequency. This is known as free oscillation. Forced oscillation occurs when a periodic driving force is applied to an object, which causes it to oscillate at a particular frequency. Resonance When the driving frequency of the external force applied to an object is the same as the natural frequency of the object, resonance occurs. This is when the amplitude of oscillation rapidly increases, and if there is no damping, the amplitude will continue to increase until the system fails. As damping is increased, the amplitude will decrease at all frequencies, and the maximum amplitude occurs at a lower frequency. Techniques to investigate resonance To investigate the resonance of an object experimentally, a mass can be suspended between two springs attached to an oscillation generator. A millimetre ruler can be placed parallel with the spring-mass system to record the amplitude. The driver frequency of the generator is slowly increased from zero, so the mass will oscillate with increasing amplitude, reaching maximum amplitude when the driver frequency reaches the natural frequency of the system. The amplitude of oscillation will then decrease again as frequency is increased further. The spring-mass system experiences damping from the air so the amplitude should not continue to increase until the point of system failure. To increase accuracy, the system can be filmed and the amplitude value recorded from video stills, as it can be difficult to determine this whilst the mass is oscillating. www.pmt.education