Define surface energy of the liquid. Obtain the relation between the surface energy and surface tension.
Surface energy is the energy needed to increase the surface area of a liquid by one unit area. It is associated with the force acting at the liquid's surface that resists expansion. Surface tension, which is the force per unit length acting along the surface, is directly related to surface energy. The relationship between surface energy \( E \) and surface tension \( T \) is expressed as: \[ E = T \cdot L \] where \( L \) is the length of the line along the surface.
Two soap bubbles of radius 2 cm and 4 cm, respectively, are in contact with each other. The radius of curvature of the common surface, in cm, is _______________.
The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.
Solve the following L.P.P. by graphical method:
Maximize:
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]