HARMONIC OCSILLATIONS M Theory 1. The period T (sec) of a simple harmonic oscillator is given by Т% 2л here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg). Frequency f (Hz) of a simple harmonic oscillator 2. f = 3. Circular frequency of oscillations wo (rad/sec) here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg) Relationship between circular frequency and period 4. 2л T =- here T is the oscillation period (sec), wo is circular frequency (rad/sec) 5. Law of harmonic motion In general, a particle moving along the x axis exhibits simple harmonic motion when x, the particle's displacement from equilibrium, varies in time according to the relationship x (t) 3 Асos(0ot + Ф), here x is the particles displacement (m) at moment of time t (sec); A is maximum displacement from equilibrium or amplitude (m); p = wot + Po is phase of oscillatory motion (rad); 4o is initial phase (rad); wo is circular frequancy (rad/sec). Position х X- Amplitude X -X Speed v (m/sec) of a simple harmonic oscillator 6. dx v(t) = = -Awosin(@,t + Po) = -vmSin(@ot + Po), here vm = Awo is maximum speed, or speed amplitude. Acceleration a (m/sec²) of a simple harmonic oscillator d?x 7. = -Awžcos(wot + 4o) = -amcos(wot + Po), here am = Aw is maximum acceleration or amplidude of acceleration. a(t) = dt2 8. Kinetic energy E (J) of a simple harmonic oscillator kA? sin²(wot + Po). ту? 2 9. Potential energy E, (J) of a simple harmonic oscillator kx2 E, = = kA? - cos²(wot + Po). %3D 10. Full energy E (J) of a simple harmonic oscillator тозА? kA² E = Ex + Ep %3D T rad =180° or 1 rad = 180° 30° G 60° () 90° G) 45° ) 0° 1 VZ/2 V3/2 1/2 sin(0) V3/2 V2/2 cos(0) 1/2 1 V3/3 V3 tg(0) cos(0 + n) = -cos® sin(0 + n) = -sin0 3. An object having mass m = 5 gr oscillates with simple harmonic motion along the x axis. Its displacement from the origin varies with time according to the equation x = 4cos (nt +- where t is in seconds and the angles in the parentheses are in radians. (a) Determine kinetic and potential energy of the object at moment of time t = 1 sec.

4.A simple pendulum has an oscillation frequency of 0.5 Hz, and a mass of 200 grams attached. At its greatest amplitude, the string forms an angle of 30 degrees. A) What is the greatest value of gravitational potential energy obtained by the mass? B) What is the fastest speed obtained by the mass?