The potential energy of a particle of mass $$4 \mathrm{~kg}$$ in motion along the x-axis is given by $$\mathrm{U}=4(1-\cos 4 x)$$ J. The time period of the particle for small oscillation $$(\sin \theta \simeq \theta)$$ is $$\left(\frac{\pi}{K}\right) s$$. The value of $$\mathrm{K}$$ is _________.

A mass $$0.9 \mathrm{~kg}$$, attached to a horizontal spring, executes SHM with an amplitude $$\mathrm{A}_{1}$$. When this mass passes through its mean position, then a smaller mass of $$124 \mathrm{~g}$$ is placed over it and both masses move together with amplitude $$A_{2}$$. If the ratio $$\frac{A_{1}}{A_{2}}$$ is $$\frac{\alpha}{\alpha-1}$$, then the value of $$\alpha$$ will be ___________.

As per given figures, two springs of spring constants $$k$$ and $$2 k$$ are connected to mass $$m$$. If the period of oscillation in figure (a) is $$3 \mathrm{s}$$, then the period of oscillation in figure (b) will be $$\sqrt{x}~ s$$. The value of $$x$$ is ___________.

A body is performing simple harmonic with an amplitude of 10 cm. The velocity of the body was tripled by air jet when it is at 5 cm from its mean position. The new amplitude of vibration is $$\sqrt{x}$$ cm. The value of x is _____________.