15 - 15.1 Simple Harmonic Motion Any motion that repeats itself at regular intervals is called harmonic motion A particle experiences a simple

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15.1. Simple Harmonic Motion Any motion that repeats itself at regular intervals is called harmonic motion . A particle experiences a simple harmonics motion if its displacement from the origin as function of time is given by where x m , [omega] and [phi] are constants, independent of time. The quantity x m is called the amplitude of the motion and is the maximum displacement of the mass. The time-varying quantity ([omega]t + [phi]) is called the phase of the motion and [phi] is called the phase constant . The phase constant is determined by the initial conditions. The angular frequency [omega] is a characteristic of the system, and does not depend on the initial conditions. The unit of angular frequency is rad/s. The period T of the motion is defined as the time required to complete one oscillation. Therefore, the displacement x(t) must return to its initial value after one period x(t) = x(t + T) This is equivalent to Using the relation it is immediately clear that The number of oscillations carried out per second is called the frequency of the oscillation . The symbol for frequency is [nu] and its unit is the Hertz (Hz): 1 Hz = 1 oscillation per second = 1 s -1 The period T and the frequency [nu] are related as follows The velocity of an object carrying out simple harmonic motion can be calculated easily The positive quantity [omega] x m is called the velocity amplitude and is the maximum velocity of the object. Note that the phase of the velocity and displacement differ by 90deg. . This means that the velocity is greatest when the displacement is zero and vice versa . The acceleration of an object carrying out simple harmonic motion is given by The positive quantity [omega] 2 x m is the acceleration amplitude a m . Using the expression for x(t), the expression for a(t) can be rewritten as This shows that the acceleration is proportional to the displacement, but opposite in sign. The force acting on the mass can be calculated using Newton's second law This equation of force is similar to the force exerted by a spring (Hooke's law)
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15 - 15.1 Simple Harmonic Motion Any motion that repeats itself at regular intervals is called harmonic motion A particle experiences a simple

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