Question

Two springs of force constants $k$ and $2k$ are connected to a mass as shown below. The frequency of oscillation of the mass is

• A. $\frac{1}{2\pi}\sqrt{\frac{k}{m}}$
• B. $\frac{1}{2\pi}\sqrt{\frac{2k}{m}}$
• C. $\frac{1}{2\pi}\sqrt{\frac{3k}{m}}$
• D. $\frac{1}{2\pi}\sqrt{\frac{m}{k}}$

Answer 1

Answer: (C)

$\frac{1}{2\pi}\sqrt{\frac{3k}{m}}$

Answer 2

When the oscillating mass $m$ is at a distance $x$ towards right from its equilibrium position, then the spring is stretched through distance $x$ while the other spring is compressed through the same distance $x$. Hence, restoring force exerted by each spring on mass $m$ is in the same direction tending to bring it in its equilibrium position. Let $F_1$ and $F_2$ be the restoring forces produced then $F_1-kx \,$ and $\, F_2 =-2kx$ Total restoring force is $F=F_1+F_2 =-kx-2kx=-(3k)x.$ Hence, frequency $n=\frac{1}{2\pi}\sqrt{\frac{3k}{m}}$