Determine the time period of the oscillation of mass m and the...

Determine the time period of the oscillation of mass

m

and the equivalent spring constant of the pair of springs for the rightmost arrangement shown in the figure.
Determine the time period of the oscillation of mass m and the equivalent spring constant of the pair of springs for

Determine the time period of the oscillation of mass m and the equivalent spring constant of the pair of springs for

Options

A $\pi \sqrt{\dfrac{m(k_1 + k_2)}{k_1 k_2}}$ and $\dfrac{k_1 k_2}{k_1 + k_2}$
B $2\pi \sqrt{\dfrac{m|k_1 - k_2|}{k_1 k_2}}$ and $\dfrac{k_1 k_2}{|k_1 - k_2|}$
C $2\pi \sqrt{\dfrac{m(k_1 + k_2)}{k_1 k_2}}$ and $\dfrac{k_1 k_2}{k_1 + k_2}$
D $2\pi \sqrt{\dfrac{m}{k_1 + k_2}}$ and $k_1 + k_2$

Correct answer

C. $2\pi \sqrt{\dfrac{m(k_1 + k_2)}{k_1 k_2}}$ and $\dfrac{k_1 k_2}{k_1 + k_2}$

Step-by-step solution

In the rightmost arrangement (c), the two springs with spring constants

k_1

and

k_2

are connected in series.

For springs in series, the equivalent spring constant

k_{eq}

is given by:

\dfrac{1}{k_{eq}} = \dfrac{1}{k_1} + \dfrac{1}{k_2}

k_{eq} = \dfrac{k_1 k_2}{k_1 + k_2}

The time period

T

of oscillation for a mass

m

attached to a spring system is:

T = 2\pi \sqrt{\dfrac{m}{k_{eq}}}

Substituting the value of

k_{eq}

T = 2\pi \sqrt{\dfrac{m}{\dfrac{k_1 k_2}{k_1 + k_2}}} = 2\pi \sqrt{\dfrac{m(k_1 + k_2)}{k_1 k_2}}

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