A spherical soap bubble of radius 3 cm is formed inside another spherical soap bubble of radius 6 cm. If the internal pressure of the smaller bubble of radius 3 cm in the above system is equal to the internal pressure of the another single soap bubble of radius r cm. The value of r is ___________.

A square aluminum (shear modulus is $$25 \times 10^{9}\, \mathrm{Nm}^{-2}$$) slab of side $$60 \mathrm{~cm}$$ and thickness $$15 \mathrm{~cm}$$ is subjected to a shearing force (on its narrow face) of $$18.0 \times 10^{4}$$ $$\mathrm{N}$$. The lower edge is riveted to the floor. The displacement of the upper edge is ____________ $$\mu$$m.

A uniform heavy rod of mass $$20 \mathrm{~kg}$$, cross sectional area $$0.4 \mathrm{~m}^{2}$$ and length $$20 \mathrm{~m}$$ is hanging from a fixed support. Neglecting the lateral contraction, the elongation in the rod due to its own weight is $$x \times 10^{-9} \mathrm{~m}$$. The value of $$x$$ is _______________.

(Given, young modulus Y = 2 $$\times$$ 10¹¹ Nm^$$-$$2 and g = 10 ms^$$-$$2)

In an experiment to determine the Young's modulus of wire of a length exactly $$1 \mathrm{~m}$$, the extension in the length of the wire is measured as $$0.4 \mathrm{~mm}$$ with an uncertainty of $$\pm\, 0.02 \mathrm{~mm}$$ when a load of $$1 \mathrm{~kg}$$ is applied. The diameter of the wire is measured as $$0.4 \mathrm{~mm}$$ with an uncertainty of $$\pm \,0.01 \mathrm{~mm}$$. The error in the measurement of Young's modulus $$(\Delta \mathrm{Y})$$ is found to be $$x \times 10^{10}\, \mathrm{Nm}^{-2}$$. The value of $$x$$ is _________________. $$\left(\right.$$take $$\mathrm{g}=10 \mathrm{~ms}^{-2}$$ )