16.1 Simple Harmonic Motion Chapter 16 OSCILLATIONS 16.1 Simple Harmonic Motion 16.1.1 Linear restoring force Linear restoring force: The magnitude of force is proportional to displacement respect to equilibrium position with opposite direction of displacement. Linear restoring torque may be defined in the same way.

16.1.2 Simple harmonic motion Spring oscillation analysis:

The equation can be solved as: We prefer using the form of cosine function Angular frequency: Amplitude: A Phase constant: 

The torsional oscillator For small twists, From rotational law: The solution is

A simple pendulum For small angle swings We obtain The solution is

The physical pendulum For small angular displacement We get where

The definition of simple harmonic motion When the resultant force or resultant torque exerted on a body is of linear restoring, the body undergoes a oscillation called simple harmonic motion. When a oscillation is described by a cosine function, the oscillation is called simple harmonic motion. To determine a SHM, three factors are needed, that is, the angular frequency, the amplitude, the phase constant.

16.1.3 Period, frequency, and angular frequency Period is a time when one complete oscillation undergoing. Frequency is numbers of oscillation in unit time.

16.1.4 Amplitude, phase, and phase constant Since We get Amplitude A is the maximum distance of an oscillator from its equilibrium position. Phase: Phase constant (or initial phase):

: angular frequency determined by the oscillation system; Amplitude A and phase constant  are determined from the initial conditions: We get Therefore,

16.1.5 Oscillation diagram

16.2 Uniform Circular Motion and SHM 16.2.1 Rotational vector

16.2.2 Phase difference Compare the phase difference of two oscillations with: If 2  1 > 0, SHM-2 is in before SHM-1; If 2  1 < 0, SHM-2 is in after SHM-1; If 2  1 = 0, SHM-2 is in synchronization with SHM-1 (or in synchronous phase); If 2  1 =  , SHM-2 is in anti-phase with SHM-1.

16.2.3 x, v, and a in SHM

16.3 Oscillation Energy The kinetic energy of a mass in simple harmonic motion The potential energy associated with the force  kx The total energy The total energy is conserved for all SHMs.

By means of total energy

16.4 The Composition of SHM 16.4.1 Two SHMs with the same direction and frequency where

DISCUSSION: If two SHMs are synchronous, If two SHMs are in antiphase, In general case,

16.4.2 Two SHMs with the same direction but different frequency : modulating frequency : average frequency

If two component frequencies are very close and their difference is small. Therefore, the average frequency is much larger than modulating frequency. The phenomenon that the composite amplitude will change periodically is named a beat.

16.4.3 Two SHMs with the same frequency but perpendicular directions If , If , If ,

16.4.4 Two SHMs with different directions and frequencies Generally, the composite motion is rather complicated and sometimes is hard to get a periodical oscillation. However, if two frequencies have a ratio of two simple integers, we still get a periodical motion of which the trajectory is called a Lissajous figure.

16.5 Damped Harmonic Motion A frictional force may be expressed as The Newton’s equation

Three situations: (i) For a small values of  , satisfying where

(ii) If damping is large, (iii) Critically damped motion:

16.6 Forced Harmonic Motion: Resonance 16.6.1 Characterization of forced oscillation

16.6.2 Resonance If , the resonant amplitude This is called a displacement resonance.

Problems: 16-26 (on page 367), 16-28, 16-36, 16-46, 16-58, 16-60,

7. A cart consists of a body and four wheels on frictionless axles 7. A cart consists of a body and four wheels on frictionless axles. The body has a mass m. The wheels are uniform disks of mass M and radius R. The cart rolls, without slipping, back and forth on a horizontal plane under the influence of a spring attached to one end of the cart. The spring constant is k. Taking into account the moment of inertia of the wheels, find a formula for the frequency of the back-and-forth motion of the cart.

8. Assume a particle joins two perpendicular SHMs of Draw the trajectory of combined oscillation of the particle by using rotational vector method.