1. 16.105 Physics I : Mechanics Motion of particles described by vectors r(x,y,z,t) = x(t) î + y(t)  + z(t) k Motion is the sum of motion along three perpendicular directions Point-like particle moves according to Newton’s Laws F = mdp/dt =m a

2. 16.107 Physics II : Waves and Modern Physics Oscillations: building blocks are periodic functions  (x,t) =  n a n cos( k n x) cos(  n t) Motion is the sum of periodic functions.  (x,t) is a solution of Schrödinger equation -(  2 /2m)  2  (x,t)/  2 x + V(x)  (x,t)= i    (x,t)/  t Solutions are wave-like

3. Chapter 17 Oscillations Oscillations are everywhere vibrations in your eardrum, cell phone, CD player, quartz watches, heart beat some of these vibrations are obvious to us and some are not even detectable however, the mathematical description is basically the same

4. Oscillations Some oscillations occur in a medium such as sound waves or water waves light, radio waves, x-rays are also oscillatory phenomena but do not involve motion of a medium but rather electric and magnetic fields sound waves need a medium light waves do not need a medium

5. Oscillations Vibrations do not continue forever unless we continually pump the system Pendulumclock  spring Quartz watch  battery we need to balance losses due to damping or friction energy is converted into heat and lost eg. slinky

6. ‘Snapshots’ of oscillating system frequency f = # of oscillations each second 1 hertz = 1 Hz = 1 s -1 period T = 1/f is the time for one complete oscillation Mass

7. Simple Harmonic Motion x m is the amplitude (maximum value)  is the phase angle (determines x(0))“phi”  is ? “omega” x(t + T) = x(t) ------>  (t+T)+  =  t +   (t+T) =  t + 2  ---->  T = 2   = 2  /T = 2  f angular frequency rads/sec recall circular motion  =  /  t x(t) = x m cos(  t +  )

8. Periodic motion or Harmonic motion Motion is repeated at regular intervals x(t) = x m cos(  t +  ) simple harmonic motion x m (positive constant) is the amplitude  t +  is the phase of the motion  is the phase constant(or phase angle) x(0) = x m cos(  )

9. Both curves have the same period and phase constant

10. Both curves have the same amplitude and phase constant

11.  t +  is the phase x(t)= x m cos(  t +  ) points with the same phase have the same value of x a decrease in  shifts the red curve to the right Both curves have the same amplitude and period x m occurs when phase  t+  =0

12. SHM x m is a constant  is a constant “phi”  is a constant “omega”  t +  varies with time t x(t) varies with time t let’s try some examples using maple x(t) = x m cos(  t +  ) shm.mws

13. Velocity v(t) = -  x m sin(  t +  ) But cos(A+B)= cos(A)cos(B)-sin(A)sin(B) Hence cos(  t +  +  /2) = - sin(  t +  ) v(t) =  x m cos(  t +  +  /2) phase constant increased --> shift to left x(t) = x m cos(  t +  )

14. Acceleration v(t) =  x m cos(  t +  +  /2) Calculate acceleration a(t)= -  2 x m cos(  t +  ) a(t)=  2 x m cos(  t +  +  /2 +  /2) phase constant increased again --> shift left x(t) = x m cos(  t +  ) v(t) = -  x m sin(  t +  )