Mathematical Physics & Mechanics

Scalar and vector fields. Derivatives of vector functions (physical examples - velocity, centripetal acceleration of a point in circular motion). Gradient of a scalar field (example of Newton’s gravitational force as gradient of a scalar potential). Gradient as normal vector to a surface. Divergence and curl of a vector field- solenoidal and irrotational vector fields. Laplacian operator (physical problems –Laplacian of gravitational potential, divergence of central force). Vector identities.

Vector integration- Line integral (physical example- work done by a force, path dependence/independence and concept of conservative force). Surface and volume integrals. Concept of vector flux. Gauss’s divergence theorem and Stokes’s theorem (statement only).

Introduction to curvilinear coordinates. Orthogonal curvilinear coordinates. Examples of spherical, cylindrical and plane polar coordinates. Line element- transformation from Cartesian to curvilinear coordinates (spherical and cylindrical). Gradient, divergence and curl in spherical and cylindrical coordinates.

Definition and properties of Dirac delta function. Representation of delta function by Gaussian function, rectangular function and Laplacian of 1/r.

Inertial frames. Non-inertial frames and fictitious forces. Uniformly rotating frame. Laws of physics in rotating coordinate systems. Centrifugal force. Coriolis force and its applications.

Motion under central force. Two-body problem and its reduction to one body problem. Kepler’s laws, Gravitational potential and fields due to spherical body. Gauss’s law and Poisson’s equation for gravitational field.

Work and kinetic energy theorem. Conservative and non-conservative forces. Potential energy. Force as gradient of potential energy. Work and potential energy. Work done by non-conservative forces. Dynamics of a system of particles. Centre of mass. Principle of conservation of momentum. Torque. Impulse. Elastic and inelastic collisions between particles. Centre of mass and laboratory frames.

Rigid body motion. Rotational motion. Moment of inertia of rectangular lamina, disc, cylindrical and spherical bodies. Kinetic energy of rotation. Motion involving both translation and rotation.

Relation between elastic constants. Twisting torque on a cylinder or wire. Cantilever. Kinematics of moving fluids: Poiseuille’s equation for flow of a liquid through a capillary tube.

• To study the motion of spring and calculate (a) spring constant and (b) rigidity modulus.
• To determine the moment of inertia of a cylinder about two different axes of symmetry by torsional oscillation method.
• To determine coefficient of viscosity of water by capillary flow method (Poiseuille’s method).
• To determine the Young’s modulus of the material of a wire by Searle’s apparatus.
• To determine the modulus of rigidity of a wire (static method).
• To determine the value of g using bar pendulum.
• To determine the value of g using Kater’s pendulum.
• To determine the height of a building using a sextant.
• To determine g and velocity for a freely falling body using digital timing technique.