Question

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.

Answer 1

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

`K=K_1+K_2`

We know that natural frequency given as

`\omega =\sqrt{(K)/(m)}`

So

`\omega =\sqrt{(K_1+K_2)/(m)}`

For series connection:

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

`(1)/(K)=(1)/(K_1)+(1)/(K_2)`

`K=(K_1K_2)/(K_1+K_2)`

Now by putting values

`\omega =\sqrt{(K)/(m)}`

`\omega =\sqrt{(K)/(m)}`

`\omega =\sqrt{(K_1K_2)/(m(K_1+K_2))}`