1. SHM NFLS Dipont A-level Physics © Adam Gibson

2. Simple Harmonic Motion Definition: If the acceleration of a body is directly proportional to its distance from a fixed point and is always directed towards that point, the motion is simple harmonic. x

3. Simple Harmonic Motion O N P Prove that if P performs circular motion with angular velocity ω, then N performs simple harmonic motion.

4. Simple Harmonic Motion O N P Starting from θ=0, sketch a graph of x against t, and v against t for a whole oscillations.

5. Simple Harmonic Motion Look at these oscillations: Ruler twanging Pendulum Atoms in a lattice Mass on a spring Plucked guitar string (after plucking!) Building swaying in an earthquake Are they free oscillations? Are they simple harmonic motion?

6. Equations of Simple Harmonic Motion Define these terms: Amplitude, period, frequency, phase difference What are the equations of simple harmonic motion? Note: you don’t need to differentiate or integrate.. 

7. Is this Simple Harmonic Motion? 0 Springs 1 and 2 have natural lengths l1 and l2, and spring constants k1 and k2 Newtons per metre. Start by finding an expression for the equilibrium position of the mass, x0.

8. Is this Simple Harmonic Motion? … or equivalent Now, by considering the equation of motion when the position of the mass is displaced from x0 to x’, prove that the mass’ motion is simple harmonic. Now use the previous equation - substitute

9. Is this Simple Harmonic Motion? So, acceleration is proportional to displacement, which is the definition of SHM. QED. Express the period of oscillations in terms of the physical constants of the system

10. A quantitative example of the double spring system Let’s analyse the motion of a real physical system: … all in SI units. Assume the mass passes the equilibrium point with a kinetic energy of 0.1J, travelling towards the right, at time t = 0. Draw a graph for one complete oscillation of: Displacement Velocity Potential Energy

11. Is this Simple Harmonic Motion? First, let’s find the maximum displacement of the mass from equilibrium: See p. 234 of AS/A2 book. This is the amplitude of the oscillation. What is the frequency of the oscillation?

12. Is this Simple Harmonic Motion? So what is x as a function of t? First, it must be a sine wave because it is zero at t=0 Second, we know the angular frequency is 10root3. Third, we know the amplitude is (5root3)^-1 What is the velocity as a function of time?

13. Is this Simple Harmonic Motion? Now find the kinetic energy as a function of time: What is the potential energy as a function of time? Notice that the total energy does not vary with time:

14. What is the phase difference between the displacement and the velocity? 1 period = 0.36s

15. What is the phase difference between the K.E. and the P.E.?