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Solving Simple Harmonic Motion

Simple harmonic motion (shm) is defined as the motion of the simple harmonic oscillator with a periodic motion which can be neither be damped nor driven. A simple harmonic motion system force is experienced by the Hooke’s law. Hooke’s law states that the force of the system is directly proportional to the displacement x and the points in the opposite direction. The simple harmonic motion is given by the equation, 

                                              `y=A (sin (omega t))`

In this article of solving simple harmonic motion we will see some how to find the displacement of a system and now, let us see some example problems in solving simple harmonic motion (shm).


Example 1 &2 in solving simple harmonic motion (shm):


Example 1:

          Find the displacement of a system having simple harmonic motion with period 2 sec and having amplitude of 6 cm. The system has an angular frequency of 12 rad/s.

Given data:

          Period (t) = 2 sec,

          Amplitude (A) = 6 cm,

          Angular frequency (ω) = 12 rad/s.

Formula:

          `y=A (sin (omega t))`

Solution:

          Substitute the given data in the formula for simple harmonic motion (shm):

          `y=` `6xx sin (12xx6)`

            `=6xx sin (72)`

            `=6xx 0.95`

            =5.7   [cm]

          Therefore, the system has a displacement of 5.7 cm.


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Example 2:

          Find the displacement of a system having simple harmonic motion with period 4 sec and having amplitude of 3 cm. The system has an angular frequency of 14 rad/s.

Given data:

          Period (t) = 4 sec,

          Amplitude (A) = 3 cm,

          Angular frequency (ω) = 14 rad/s.

Formula:

          `y=A sin (omega t)`

Solution:

          Substitute the given data in the formula for simple harmonic motion (shm):

          `y=4xx sin (14xx4)`

            `=4xx sin (56)`

            `=4xx 0.82`

            =3.28   [cm]

          Therefore, the system has a displacement of 3.28 cm.


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Example 3 &4 in solving simple harmonic motion (shm):


Example 3:

          Find the displacement of a system having simple harmonic motion with frequency 5 Hz and having amplitude of 3 cm. The system has an angular frequency of 10 rad/s.

Given data:

          frequency (f) = 5 Hz,

          Amplitude (A) = 3 cm,

          Angular frequency (ω) = 10 rad/s.

Formula:

          `y=A sin (omega t)`

Solution:

          First we need to find the period (t),

          In the question the frequency is given, therefore:

                   t=" `1/f`

                   t=" `1/5`

                   t="0.2" [sec]

          Substitute the given data in the formula for simple harmonic motion (shm):

          `y=3xx sin (10xx0.2)`

            `=3xx sin (2)`

            `=3xx 0.034`

            =0.102   [cm]

          Therefore, the system has a displacement of 0.102 cm..

Example 4:

Find the displacement of a system having simple harmonic motion with frequency 4 Hz and having amplitude of 10 cm. The system has an angular frequency of 5 rad/s.

Given data:

          frequency (f) = 4 Hz,

          Amplitude (A) = 10 cm,

          Angular frequency (ω) = 5 rad/s.

Formula:

         ` y="A" sin (omega t)`

Solution:

          First we need to find the period (t),

          In the question the frequency is given, therefore:

                   t=" `1/f`

                   t=" `1/4`

                   t="0.25" [sec]

          Substitute the given data in the formula for simple harmonic motion (shm):

          `y=10xx sin (5xx0.25)`

            `=10xx sin (1.25)`

            `=10xx 0.021`

            =0.21   [cm]

          Therefore, the system has a displacement of 0.21 cm.