Solve your IGNOU Doubts

Question:

Two objects of equal mass are connected by a three spring with spring constants as shown in figure. Obtain the normal mode frequencies.

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Question:

The orbit of a particle in a central force field is given by $r = k e^{\alpha \theta}$ , obtain the force law.

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Question:

Locate and name the singularity of the function:

$$\frac{z^2 - 2z}{z^2 + 2z + 2}$

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Question:

Obtain the analytic function whose real part is

$$u(x, y) = e^x \cos y$

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Question:

A central potential is given as V(r):

(i) Obtain the criterion for a stable circular orbit.

(ii) If the central potential is given as $V(r) = -k/r^n, n \geq 1$ . For what values of n are the circular orbits stable.

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Question:

Define covariant and cotravariant tensors of rank 2. Prove that $a_i = g_{ij}v^j$ transform covariantly, where g_ij are the components of the matrix tensor of rank 2 and vⁱ the components of a contravariant vector.

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Question:

Obtain the eigenvalues and eigenvectors of matrix A:

$$A = \begin{bmatrix} 2 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 2 \end{bmatrix}$

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Question:

The first Pauli matrix is

$$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
calculate $U_1(\theta) = \exp(i\theta \sigma_1) = 1 + i\theta \sigma_1 - \frac{\theta^2}{2} \sigma_1^2 - \dots$
For real $\theta$ , show that U₁ is unitary and has determinant

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Question:

Consider a simple pendulum of mass m whose length changes with time as $l = l_0 t^2, (x^2 + y^2 = l_0^2 t^4)$ . Obtain the Lagrange equation of motion, the generalised momentum and the energy function for the system.

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Question:

Show that the following vectors

$$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$

are linearly independent.

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Question:

Obtain orthogonality relation for Hermite polynomials using the generating function:

$$e^{2xt-t^2} = \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!}$

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Question:

Using the generating function for Bessel functions of the first kind and integral order

$$g(x, t) = \exp \left[ \frac{x}{2} \left( t - \frac{1}{t} \right) \right] = \sum_{n=-\infty}^{\infty} J_n(x) t^n$

Obtain the recurrence relation

$$J_{n-1}(x) + J_{n+1}(x) = 2 \frac{n}{x} J_n(x)$

Also using the generating function show that

$$J_0(x) + 2J_2(x) + 2J_4(x) + 2J_6(x) + \dots + 2J_{2k}(x) + \dots = 1$

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Question:

A pendulum of mass m is pivoted to a support of mass M sliding without friction along x-axis as shown in the figure. Obtain the Lagrange equation of motion for the given system.

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Question:

Use the method of separation of variables to reduce the Laplace's equation $\nabla^2 f = 0$ into three ODEs.

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Question:

Two equal of mass m attached by a thread of length b are oriented vertically as shown in the figure. The acceleration decrease according to height as $g = g_0(1 - z / a)$ , where g₀ and a are constants. Calculate the vertical acceleration of the system as measured from the centre of the thread.

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Question:

Derive an integral equation corresponding to the ODE:

$$y'' - 2y = 0$
subject to the conditions: $y(0) = 4; y'(0) = -2$

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Question:

Reduce the following PDE into three ODEs:

$$\left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right] f(x, y, z) + k^2 f(x, y, z) = 0$

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Question:

Two frictionless blocks of equal masses m are connected by a massless rigid rod of length L. The system is constrained to move in the vertical plane under the action of an applied force $\vec{F}$ as shown in the figure. Obtain the equation of motion of the system and calculate the condition for static equilibrium.

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Question:

What is meant by magnetic reluctance? Derive an expression for the reluctance of a composite magnetic circuit consisting of an iron core and an air gap. Clearly state the assumptions made.

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Question:

A long straight wire carries a steady current I. The wire is bent to form a circular arc of angle $\theta$ and radius R. Calculate the magnetic field at the centre of curvature and determine its direction using the right-hand rule.

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