Physics 141Mechanics Lecture 21 Oscillation Yongli Gao You may not know it, but every atom/molecule in your body is oscillating. For any system, there's at least one state that the system is of the lowest potential energy. This is a point of stable equilibrium, or the bottom of the valley in a potential vs. position curve. If the system is of a small displacement from the point, it'll experience a restoring force, pointing to the bottom of the potential curve. The force accelerates the system so it'll swing across the equilibrium point to the other side, and the restoring force will reverse as well. Thus, it'll oscillate around the point of equilibrium.
Period and Frequency The period T of an oscillation is the time taken for the oscillating system to repeat itself, or, to complete one oscillation. For example, the time for a swinging pendulum starting from one extreme point to the come to the same point. Same position, velocity, and acceleration. T is in second. The frequency f of an oscillation is the number of complete oscillations per unit time. Clearly The unit for frequency is hertz. 1 hertz = 1/s
Simple Harmonic Oscillator The simplest oscillation is a particle of mass m attached to a massless spring of spring constant k on a horizontal frictionless plane. From Hooke's law, F=-kx From Newton's 2nd law, The solution is the simple harmonic oscillation (SHO) where A is the amplitude of the oscillation, is the angular frequency, t+ the phase, and the phase constant. The period of an SHO is Any periodic motion can be expressed as the sum of SHO's of different frequencies.
Energy of an SHO In an SHO, the kinetic energy K and potential energy U convert to each other back and forth, but the total energy E=U+K is a constant.
General Small Oscillations Any small oscillation about an equilibrium position can be approximated as SHO. Suppose the the potential energy is U(x) and the equilibrium position is x 0. At equilibrium the force is zero, Taylor expansion
Circular Reference SHO motion can be viewed as the x-component of a uniform circular motion. x y r tt
Simple Pendulum A simple pendulum is formed by hanging a particle of mass m to a pivotal point O by a massless string of length l. About O, This has been used in the past centuries for clocks.
Physical Pendulum A physical pendulum is formed by allowing a rigid body fixed to a pivotal point O to oscillate frictionlessly. About O, This is true for real pendulums.
Damped Oscillation Real objects may experience friction or viscosity as they oscillate. The motion is damped oscillation. Viscous force The amplitude is damped, and the energy dissipates as
Forced Oscillation You can also drive an object to oscillation by applying a periodic force, like walking on a hanging bridge. The amplitude depends on both =√k/m and F. If the driving frequency F is the same as the natural frequency , the amplitude reaches the maximum and we have resonance.