Pressure inside the bubble in vacuum \(\rho = \frac{4S}{r}\)
Volume of bubble, V= \(\frac{4}{3} \pi r^{3}\)
as PV = a constant
\(\frac{4s}{R}\times \frac{4}{3} \pi r^{3}=\frac{4S}{r_{1}}\times \frac{4}{3} \pi r^{3}_{1}+\frac{4S}{r_{2}}\times \frac{4}{3}\pi r^{3}_{2}\)
Or \(R^{2}=r^{2}_{1}+r^{2}_{2}\) 
= 3²+4² = 25
R=5 cm

\(\Delta l=x ; y =\frac{F/A}{\Delta l/L}\ or \ F=\frac{yA \Delta l}{L}\)
The work done from 0 to x = Δl change in length.
Average distance = \(\frac{0+\Delta l}{2}=\frac{\Delta l}{2}\)
work done = force \(\times\)distance
\(= \frac{yA \Delta l}{2} \times \frac{\Delta l}{2}=\frac{ya (\Delta l)^{2}}{2L}\)
=\(\frac{yA x^{2}}{2L}\)

No, because stress is a scalar quantity, not a vector quantity.
Stress = \(\frac{Magnitude \ of\ internal\ reaction \ force}{Area \ of\ cross-section}\)

(i) Steam at high pressure is made to enter the, cylinder of vacuum brake.
(ii) Due to high velocity, pressure decreases in accordance with Bernoulli's principle.
(iii) Due to this decrease in pressure, the piston gets lifted. Consequently brake gets lifted.

When we suck through the straw, the pressure inside the straw becomes less than the atmospheric pressure. Due to the pressure different the soft drink rises in the straw and we are able to take the soft drink easily.

(i) We know that atmospheric pressure decreases with height. Since ink inside the pen is filled at the atmospheric pressure existing on the surface of earth, it tends to come out to equalise the pressure.
(ii) This can spoil the clothes of the passengers, so they are advised to remove the ink from the pen.

When we cut vegetables with a sharp knife the area of contact is less. Hence the stress applied becomes more \([Stress =\frac{\text { Force }}{\text { Area }}=\frac{F}{\Delta A}]\) the vegetables would be cut easily. So, it is so easy to cut vegetables with a sharp knife rather than a blunt knife.

Surface tension of a liquid is defined as the force of tension acting on a unit length of an imaginary line drawn on the free surface of the liquid, the direction of force being perpendicular to the line so drawn and acting parallel to the surface.
Unit is Nm^-1 and dimension is MT^-1

If two streamlines cross each other, then there will be two directions of flow at the point of intersection which is impossible.
Hence two streamlines cannot cross each other.

According to Bernoulli's theorem, the sum of pressure energy, kinetic energy, and potential energy per unit mass of an incompressible, nonviscous fluid in a streamlined flow remains a constant.    

Poisson's ratio is defined as the ratio of relative contraction (lateral strain) to relative expansion (longitudinal strain).

Hooke's law of elasticity states that within the elastic limit, the strain produced in a body is directly proportional to the stress applied.
strain ∝ stress

Stress: Stress is defined as the restoring force per unit area.
Stress, \(\sigma=\frac{Force}{Area}\)
Strain: Strain is defined as the rates of change in size to the original size if an object.
Strain, ε \(=\frac{\Delta l}{l}\)

The energies possessed by a liquid are kinetic energy, potential energy and pressure energy.
\(\text {Kinetic energy } =\frac{1}{2} m v^{2}  \)
\(\text {Potential energy } =m g h=\rho g h  \)
\(\text {Pressure energy } =\mathrm{PV}=\frac{P}{\rho}\)

In a solid, interatomic forces bind two or more atoms together and the atoms occupy the positions of stable equilibrium. When a deforming force is applied on a body, its atoms are pulled apart or pushed closer. When the deforming force is removed, interatomic forces of attraction or repulsion restore the atoms to their equilibrium positions. If a body regains its original shape and size after the removal of deforming force, it is said to be elastic and the property is called elasticity. The force which changes the size or shape of a body is called a deforming force.
Examples: Rubber, metals, steel ropes.