1. Introduction Key features

2. Simple Harmonic Motion is a type of periodic or oscillatory motion The object moves back and forth over the same path, like a mass on a spring or a pendulum We’re interested in it because we can use it to generalise about and predict the behaviour of a variety of repetitive motions

3. Period is the time taken for the motion to repeat one cycle Frequency is the number of cycles in a second Example: a cork bobbing on water is observed to move up and down twelve times in one minute. Find the frequency and the period T = 60/12 = 5.7sf = 1/T = 1/5.7 = 0.18Hz

4. An important concept in SHM is equilibrium – this is the point in the middle of the motion Amplitude is the maximum displacement from the equilibrium Equilibrium position Amplitude – pendulum at maximum displacement

5. In SHM, we consider displacement, velocity, acceleration, force and energy

6. The key to SHM (compared to other kinds of repetitive motion) is the way that force and acceleration change during each cycle Consider these two situations. Each one shows displacement from the equilibrium. In what direction will they move when released?

7. The mass on the spring and the pendulum will both move back towards the equilibrium when released Considering forces shows us that there will be a force acting towards the equilibrium

8. The weight force and the tension force are not equal and opposite If we resolve the forces we see that there is an inward- acting unbalanced force. This is the restoring force, and it causes an inward acceleration

9. For motion to be Simple Harmonic Motion: there must be a restoring force acting towards the equilibrium the force is larger when the displacement is larger, and is a maximum at maximum displacement the force is zero at the equilibrium position

10. We can also state the conditions for SHM using acceleration For motion to be Simple Harmonic Motion: The acceleration is directly proportional to the displacement from the equilibrium position The acceleration is always directed towards the equilibrium position

11. We can express the acceleration mathematically a = - c y Where c is a positive constant. The value of c depends on the situation being considered

12. If we go back to the pendulum, we can consider how F and a change as the pendulum’s displacement changes As the angle Θ decreases – that is, as the displacement from the equilibrium decreases – you can see that the restoring force will get smaller We’ve already considered the position of maximum displacement, and how there is an inward acting unbalanced force. We can find this force mathematically as F r =mgsinΘ

13. Since we know that the restoring force is unbalanced, we know, via Newton’s First Law, that there will be acceleration. If the force gets smaller as the displacement gets closer to equilibrium, the acceleration must get smaller too At equilibrium, the weight force and the tension force are balanced, so the restoring force is zero and so, therefore, is the acceleration