Question

A mass of 1 kg is suspended by means of 3 springs as shown in the figure. The spring constant $k_1,k_2,\text{and}{k}_3$ are respectively 1 kN/m, 3 kN/m and 2 kN/m. The natural frequency of the system is approximately

• A. 46.90 rad/s
• B. 52.44 rad/s
• C. 60.55 rad/s
• D. 77.46 rad/s

Answer 1

The correct option is B 52.44 rad/s

$k_1\text{and}{k}_2$ are in series, therefore $k_{eq}$, will be

$\frac{1}{k_{e{q}_1}}=\frac{1}{k_1}+\frac{1}{k_2}=\frac{1}{1}+\frac{1}{3}=\frac{4}{3}$

Now, $k_{e{q}_1}$ and $k_3$ are in parallel arrangement:

$k_{eq}=k_{e{q}_1}+k_3=\frac{3}{4}+2=\frac{11}{4}\text{kN/m}$

Natural frequency
$=\sqrt{\frac{k_{eq}}{m}}=\sqrt{\frac{11\times {10}^3}{4\times 1}}=52.44\text{rad/s}$