15.1 Simple Harmonic Motion
- Periodic motion is a repeating oscillation. The time for one oscillation is the period T and the number of oscillations per unit time is the frequency f. These quantities are related by .
- Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement.
- Maximum displacement is the amplitude A. The angular frequency , period T, and frequency f of a simple harmonic oscillator are given by , , where m is the mass of the system and k is the force constant.
- Displacement as a function of time in SHM is given by.
- The velocity is given by .
- The acceleration is , where .
15.2 Energy in Simple Harmonic Motion
- The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = −kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
- Elastic potential energy U stored in the deformation of a system that can be described by Hooke’s law is given by
- Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
- The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using
15.3 Comparing Simple Harmonic Motion and Circular Motion
- A projection of uniform circular motion undergoes simple harmonic oscillation.
- Consider a circle with a radius A, moving at a constant angular speed . A point on the edge of the circle moves at a constant tangential speed of . The projection of the radius onto the x-axis is , where is the phase shift. The x-component of the tangential velocity is .
- A mass m suspended by a wire of length L and negligible mass is a simple pendulum and undergoes SHM for amplitudes less than about . The period of a simple pendulum is , where L is the length of the string and g is the acceleration due to gravity.
- The period of a physical pendulum can be found if the moment of inertia is known. The length between the point of rotation and the center of mass is L.
- The period of a torsional pendulum can be found if the moment of inertia and torsion constant are known.
15.5 Damped Oscillations
- Damped harmonic oscillators have non-conservative forces that dissipate their energy.
- Critical damping returns the system to equilibrium as fast as possible without overshooting.
- An underdamped system will oscillate through the equilibrium position.
- An overdamped system moves more slowly toward equilibrium than one that is critically damped.
15.6 Forced Oscillations
- A system’s natural frequency is the frequency at which the system oscillates if not affected by driving or damping forces.
- A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
- The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.