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Two springs having stiffness k1 and k2, respectively, are used in a vertical position to support a single object of mass m. Show that the angular frequency of oscillation is [(k1 + k2)/m]"2 if the springs are tied in parallel, and [k1k2/(k1 + k2)m]"2 if the springs are tied in series.Two springs having stiffness k1 and k2, respectively, are used in a vertical position to support a single object of mass m. Show that the angular frequency of oscillation is [(k1 + k2)/m]"2 if the springs are tied in parallel, and [k1k2/(k1 + k2)m]"2 if the springs are tied in series.

Respuesta :

Answer:

Stiffness of spring 1 =K1

Stiffness of spring 2=K2

Mass =m

For parallel connection:

As we know that when spring are connects in parallel connection then equivalent stiffness given as

[tex]K=K_1+K_2[/tex]

We know that natural frequency given as

[tex]\omega =\sqrt{\dfrac{K}{m}}[/tex]

So

[tex]\omega =\sqrt{\dfrac{K_1+K_2}{m}}[/tex]

For series connection:

As we know that when spring are connects in series connection then equivalent stiffness given as

[tex]\dfrac{1}{K}=\dfrac{1}{K_1}+\dfrac{1}{K_2}[/tex]

[tex]K=\dfrac{K_1K_2}{K_1+K_2}[/tex]

Now by putting values

[tex]\omega =\sqrt{\dfrac{K}{m}}[/tex]

[tex]\omega =\sqrt{\dfrac{K}{m}}[/tex]

[tex]\omega =\sqrt{\dfrac{K_1K_2}{m(K_1+K_2)}}[/tex]

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