Simple harmonic motion

Displacement x(t) is a sine function of time

x(t) = A sin (2 π f t)         (when the body is at t= 0 s in a positive direction through the central position)

  • x(t)  displacement at   t = t (s)
  • A amplitude         (x en A : the same unit of length)

Example

A body is in a simple harmonic motion.  f = 20.0 Hz.    A = 5.0 cm   

At  t= 0 s is the body in a positive direction through the central position . In other words   φ(0) = 0

Find the displacement at t = 0.018 s

x = A sin (2  π f t) = 5.0 sin (2 π x 20.0 x 0.018) = 5.0 sin (2.26) = 5.0 x 0,77 = 3.85 cm   

Remark:  set calculator to  RADIANS !!!

Example

At what times (t < 0.05 s) in the above example is the displacement  x = +3.0 cm ?

Simple harmonic motion

x(t) = A sin (2 π f t)        

3.0 = 5.0 sin (2 π 20 t)                                                               

3.0/5.0 = sin (2 π 20 t)

0.6 = sin (2 π 20 t)      ( sin-1 0.6 = 0.64 )

0.64 = 2 π 20 t

t  = 5.12 x 10-3 s  

The second time  :

t= 0.025 – 5.12 x 10-3 = 0.0198 s =

19.8 x 10-3 s

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