Question

A block of mass $m$ is attached to three springs $A,B$ and $C$ having force constant $k,k$ and $2k$ respectively as shown in figure. If the block is slightly pushed against spring $C$. If the angular frequency of oscillations is $\sqrt{\frac{Nk}{m}}$, then find $N$. The system is placed on horizontal smooth surface.

Answer 1

The block is pushed against C means it is pushed towards the wall of C.

Since the push is slight, the tilt in the orientation if springs A and B can be ignored.

When the block is pushed against C, the springs A and B get pulled and they will apply to restore force.

If the block is pushed by a distance $x,$ then by resolution of vectors, the component of displacement along the line of A and B are both $\frac{x}{\sqrt{2}}.$ So the restoring force exerted by A and B will be both $\frac{kx}{\sqrt{2}}.$

By parallelogram law of vector addition, they will get added up as $\sqrt{(\frac{kx}{\sqrt{2}}{)}^2+(\frac{kx}{\sqrt{2}}{)}^2}=kx.$ The resultant restoring force kx will be along the diagonal between A and B.

Hence A and B will behave as if only one spring is attached opposite to the line of C of force constant k.

When the block is displaced towards C, both the springs will apply restoring force, hence their force constant gets added up to $3k.$

We know that angular frequency $\omega =\sqrt{\frac{k}{m}}.$

In this case it'll be $\omega =\sqrt{\frac{3k}{m}}.$

It is given that $\omega =\sqrt{\frac{Nk}{m}}.$

$\therefore N=3.$