Question 289: Oscillations

Question

Four massless springs whose force constants are $$2 \mathrm{~K}, 2 \mathrm{~K}, \mathrm{~K}$$ and $$2 \mathrm{~K}$$ respectively are attached to a mass $$\mathrm{M}$$ kept on a frictionless plane as shown in figure, If mass $$M$$ is displaced in horizontal direction then frequency of oscillating system is

Accepted Answer

On the right hand side of the block, springs are connected in parallel
$$	herefore$$ Their effective spring constant is given by
$$\begin{aligned}
& \mathrm{K}_1=\mathrm{K}+2 \mathrm{~K} \
& \mathrm{~K}_1=3 \mathrm{~K}
\end{aligned}$$
On the left hand side of the block, springs are connected in series.
$$	herefore$$ Their effective spring constant is given by,
$$\begin{aligned}
& \frac{1}{\mathrm{~K}_2}=\frac{1}{2 \mathrm{~K}}+\frac{1}{2 \mathrm{~K}} \
	herefore \quad & \mathrm{~K}_2=\mathrm{K}
\end{aligned}$$
$$	herefore \quad$$ Effective spring constant of the system is given by,
$$\begin{aligned}
& \quad \mathrm{K}_E=3 \mathrm{~K}+\mathrm{K}=4 \mathrm{~K} \
& 	herefore \quad \omega=\sqrt{\frac{\mathrm{K}_E}{M}}=\sqrt{\frac{4 K}{M}} \
& 	herefore \quad \mathrm{f}=\frac{\omega}{2 \pi}=\frac{1}{2 \pi} \sqrt{\frac{4 K}{M}}
\end{aligned}$$

Answer

$$\frac{1}{2 \pi} \sqrt{\frac{\mathrm{K}}{4 \mathrm{M}}}$$

Answer

$$\frac{1}{2 \pi} \sqrt{\frac{4 \mathrm{~K}}{\mathrm{M}}}$$

Answer

$$\frac{1}{2 \pi} \sqrt{\frac{\mathrm{K}}{7 \mathrm{M}}}$$

Answer

$$\frac{1}{2 \pi} \sqrt{\frac{7 \mathrm{~K}}{\mathrm{M}}}$$

Oscillations

Four massless springs whose force constants are $2 \mathrm{~K}, 2 \mathrm{~K}, \mathrm{~K} and 2 \mathrm{~K} respectively are attached to a mass \mathrm{M} kept on a frictionless plane as shown in figure, If mass M$ is displaced in horizontal direction then frequency of oscillating system is

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