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# Simple Harmonic Motion and waves: Velocity and acceleration

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https://brainmass.com/physics/acceleration/simple-harmonic-motion-waves-velocity-acceleration-64551

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Please do 14, 15 and 16 with step by step solutions showing equations used and answers. Please show all steps and thnaks for help.

The oscillator is completing 43 vibrations in 20.4 s hence the frequency of the oscillator i.e. the number of oscillations completed in one second is given by

n = 43/20.4 = 2.1078 Hz

The frequency of the wave is the same as that of the oscillator.

The wave velocity is the distance covered by the wave in one second. As the distance covered by the particular crest in 12.4 second is 420 cm hence the wave velocity is given by
c = 420/12.4 = 33.871 cm/s

Now as we know that the wavelength, wave velocity and the frequency is related as
c = n where  is the wavelength, hence

 = c/n = 33.871/2.1078 = 16.069cm = 0.16069 m.

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This function represents the motion of a particle of medium. When wave travels through a medium, the particles of the medium oscillate. During the oscillation the displacement of a particular particle is given by this equation.

x = A cos t.

where A is called the amplitude of oscillation and  is angular frequency.

The velocity of this particle can be obtain by differentiating above equation with respect to the time, hence

v = dx/dt = - Asin t = -3.1*27 sin(27*9.5) = - 83.7 sin(256.5rad)

= - 83.7*(-0.896) = 74.99 = 75 m/s

Here t is 256.5 rad = 81.646  rad which is equivalent to 82 - 0.354  radians
Equivalent to - 63.720 and sin(-63.720) = - 0.896

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The acceleration of the particle is given by

a = dv/dt = = - 3.1*272 cos(-63.720)
= - 2259.9*0.4427 = - 1000.4 m/s2.

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