simple harmonic motion, Lecture notes of Physics

Download simple harmonic motion and more Lecture notes Physics in PDF only on Docsity! FC312E PHYSICS Theme: 2 Hour: 1 Simple Harmonic Motion Module Learning Outcome Academic Literacy Use SHM formulae to understand oscillating objects e.g. pendulum, spring Use theoretical concepts to analyse real-world practices.  Apply in real-world contexts Develop ability to read, comprehend and express mathematical arguments Use subject-specific vocabulary effectively. Introduction to Simple Harmonic Motion  Hour 1 and 2 Simple Harmonic Motion  Hour 3 and 4 Energy in SHM  Hour 4 - 7 Worksheets  Hour 8 SHM Quiz SIMPLE HARMONIC MOTION Kaplan International Pathways Outline (Hour 1 - 2) • Simple harmonic motion – frequency, time period, mathematical representation • Pendulum • Mass on a spring Spring and Pendulum • When the period for each complete cycle of oscillation is constant, the oscillation is called simple harmonic. • In simple harmonic motion a body takes the same time for each complete oscillation. • One example of SHM is a pendulum bob oscillation. • If the angle of swing is small then its motion is approximately simple harmonic. Describing Oscillations amplitude A B • The period, T, of the oscillation is the time taken for one complete oscillation. – i.e. the time taken to travel from A to B and back again to A. • The amplitude, A, is the maximum displacement from the centre of the oscillation. – As the angle has to be small for SHM, then the amplitude must also be small. • Frequency, f, is the number of oscillations per second (which is constant) and is given by: – f=1/T with the units of Hertz, Hz Describing Oscillations Sine and Cosine waves This spray paint oscillation experiment (click on image) shows how simple harmonic oscillators create sine/cosine waves. • Let’s draw the displacement as a function of time as a cosine wave which we can draw just as we drew general waves in week 1: • It is typical to write the equation of this simple harmonic oscillation wave as: 𝒙 (𝒕 )=𝑨𝒄𝒐𝒔 (𝝎𝒕) • Notice from the graphs on the previous slide the amplitudes of the velocity and acceleration. • The amplitude of v and a will give you the maximum velocity and maximum acceleration. • Where A was the amplitude of the position x(t) for SHM. Amplitudes of v(t) and a(t) 𝒗𝒎𝒂𝒙=𝐀𝝎 𝒂𝒎𝒂𝒙=𝐀𝝎𝟐 This equation tells us quite a lot about SHM and is often used as the defining equation of SHM. The two key things it tells us about acceleration in SHM are: The key equation for SHM: This minus sign tells you that the acceleration in SHM is always in the opposite direction to the displacement Acceleration is always proportional to displacement 1 2 • The first key example of SHM is the pendulum. • A pendulum of mass, m, with arm length L, swinging through a small angle, θ has a restoring force that acts towards the equilibrium position and causes the simple harmonic motion. The Pendulum • A pendulum is constructed from a string 0.627 m long attached to a mass of 0.025 kg. When set in motion, the pendulum completes one oscillation every 1.59 s. If the pendulum is held at rest and the string is cut, how long will it take for the mass to fall through a distance of 1 m? Pendulum Example 20 • A pendulum is constructed from a string 0.627 m long attached to a mass of 0.025 kg. When set in motion, the pendulum completes one oscillation every 1.59 s. If the pendulum is held at rest and the string is cut, how long will it take for the mass to fall through a distance of 1 m? • Solution: find g using then use formula from kinetics: t = 0.452 s Pendulum Example 21 g L T 2 2 2 1 atx  • A mass oscillating on a spring is another example of SHM. • The restoring force F = kx where x is the vertical displacement of the spring from equilibrium. This is Hooke’s Law, and holds provided extension is within the elastic limit of the spring. • Typically, x is a harmonic function: x(t) = Acos(ωt) • This will balance with Newton’s 2nd Law: F = ma Mass on a Spring