Oscillations About Equilibrium CH13 Claudia 1

OSCILLATIONS ABOUT EQUILIBRIUM Chapter 13 Units of Chapter 13 • Periodic Motion • Simple Harmonic Motion • Connections between Uniform Circular Motion and Simple Harmonic Motion • The Period of a Mass on a Spring • Energy Conservation in Oscillatory Motion • The pendulum 13-1 Periodic motion Period: time required for one cycle of periodic motion Frequency: number of oscillations per unit time This unit is called the Hertz: 13-2 Simple Harmonic Motion A spring exerts a restoring force that is proportional to the displacement from equilibrium: 13-2 Simple Harmonic Motion A mass on a spring has a displacement as a function of time that is a sine or cosine curve: Here, A is called the amplitude of the motion. 13-2 Simple Harmonic Motion If we call the period of the motion T – this is the time to complete one full cycle – we can write the position as a function of time: It is then straightforward to show that the position at time t + T is the same as the position at time t, as we would expect. Connections between Uniform Circular Motion and Simple Harmonic Motion 13-3 An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion: Connections between Uniform Circular Motion and Simple Harmonic Motion 13-3 Here, the object in circular motion has an angular speed of where T is the period of motion of the object in simple harmonic motion. Connections between Uniform Circular Motion and Simple Harmonic Motion 13-3 The position as a function of time: The angular frequency: Connections between Uniform Circular Motion and Simple Harmonic Motion 13-3 The velocity as a function of time: And the acceleration: Both of these are found by taking components of the circular motion quantities. 13-4 The Period of a Mass on a Spring Since the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that Substituting the time dependencies of a and x gives 13-4 The Period of a Mass on a Spring Therefore, the period is 13-5 Energy conservation in oscillatory motion In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring: Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies: 13-5 Energy conservation in oscillatory motion As a function of time, So the total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa.