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
Units of Chapter 13 The Pendulum Damped Oscillations Driven Oscillations and Resonance
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.
13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:
13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion 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.
13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion The position as a function of time: The angular frequency:
13-3 Connections between Uniform Circular Motion and Simple Harmonic Motion 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.
13-5 Energy Conservation in Oscillatory Motion This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.
13-6 The Pendulum A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass). The angle it makes with the vertical varies with time as a sine or cosine.
13-6 The Pendulum Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).
13-6 The Pendulum However, for small angles, sin θ and θ are approximately equal.
13-6 The Pendulum Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:
13-6 The Pendulum A physical pendulum is a solid mass that oscillates around its center of mass, but cannot be modeled as a point mass suspended by a massless string. Examples:
13-6 The Pendulum In this case, it can be shown that the period depends on the moment of inertia: Substituting the moment of inertia of a point mass a distance l from the axis of rotation gives, as expected,
13-7 Damped Oscillations In most physical situations, there is a nonconservative force of some sort, which will tend to decrease the amplitude of the oscillation, and which is typically proportional to the speed: This causes the amplitude to decrease exponentially with time:
13-7 Damped Oscillations This exponential decrease is shown in the figure:
13-7 Damped Oscillations The previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.
13-8 Driven Oscillations and Resonance An oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.
13-8 Driven Oscillations and Resonance If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.
Summary of Chapter 13 Period: time required for a motion to go through a complete cycle Frequency: number of oscillations per unit time Angular frequency: Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium.
Summary of Chapter 13 The amplitude is the maximum displacement from equilibrium. Position as a function of time: Velocity as a function of time:
Summary of Chapter 13 Acceleration as a function of time: Period of a mass on a spring: Total energy in simple harmonic motion:
Summary of Chapter 13 Potential energy as a function of time: Kinetic energy as a function of time: A simple pendulum with small amplitude exhibits simple harmonic motion
Summary of Chapter 13 Period of a simple pendulum: Period of a physical pendulum:
Summary of Chapter 13 Oscillations where there is a nonconservative force are called damped. Underdamped: the amplitude decreases exponentially with time: Critically damped: no oscillations; system relaxes back to equilibrium in minimum time Overdamped: also no oscillations, but slower than critical damping
Summary of Chapter 13 An oscillating system may be driven by an external force This force may replace energy lost to friction, or may cause the amplitude to increase greatly at resonance Resonance occurs when the driving frequency is equal to the natural frequency of the system