Consider a potential V (r) = kr² , k > 0.

(i) Obtain the expression for effective potential V_eff (r) .

(ii) Determine the nature of the force associated with the potential.

(iii) Obtain r_min  and for a two-body system.

(iv) Show that the Laplace-Runge-Lenz vector is not conserved for the given potential.

Consider a spring mass pendulum, where m is the mass of the pendulum. The motion of the pendulum is constrained as given in the figure:

The motion is assume to take place only in the vertical plane. The equilibrium length of the spring is l , while its angle with the vertical is θ(t) . Obtain the equations of motion for x and θ.

Derive the Euler-Lagrange equation of motion for a bead of mass m sliding due to gravity on a circular wire.

A bead of mass m slides on a smooth rod which is rotating about one fixed end in a vertical plane with uniform angular velocity ω .

Write the Lagrangian for the system and obtain the equation of motion.

What are holonomic and non- holonomic constraints ? Write down the equations for the constraints for three point masses connected by rigid rods of length L.

Differentiate between the effects of operator crossover and mutation in the genetic algorithm with suitable example.

Consider a Kohonen self-organizing net (given below) with two cluster units and five input units. The weight vectors for the cluster units are given by

w₁ = { 3.0,5.0,7.0,9.0,0.1 }

w₂ = { 0.1,9.0,7.0,5.0,3.0 }

Use the square of the Euclidean distance to find the winning cluster unit for the input pattern (x)

x = [ 0.0  5.0  0.1  5.0  0.0 ]

Using learning rate of 0.25, find the new weights for the winning unit.

Using max-min composition, find the relation between R and S:

Design a Hopfield Network for 4-bit bipolar patterns. The training patterns are

S₁ =  [1 1 -1 -1]

S₂ = [ -1 1 -1 1]

S3 =  [ -1 -1 -1 1]

Find the weight matrix and energy for the three input samples. Determine the pattern to which the sample S = [−1 1 −1 −1]associates.

Give an example of Radial Basis Function Network and draw its network diagram.

Improve the solution of the following problem using the genetic algorithm:

Maximize  subject to 1≤ x ≤ 25 by considering the length of the string 4. Show only one iteration.

State the Schema theorem.

Consider the set of pattern vectors P. Obtain the connectivity matrix (CM) for the patterns in P (four patterns).

Describe the Function Approximation in MLP. Also, explain Generalization of MLP.

Define Kohonen networks with examples.

Consider a dataset of six points given in the following table, each of which has two features f₁ and f₂ . Assuming the values of the parameters c and m as 2 and the initial cluster centers v₁ = (5,5 ) and v₂ (10 ,10, ) , apply FCm algorithm to find the new cluster center after one iteration.

Apply the “very” hedge on the fuzzy sets defined in Q. 1(b) to get the new modified fuzzy sets. Show the modified fuzzy sets through numeration.

Construct the α − cut at α = 4.0 for the fuzzy sets defined in Q. 1(b).

Write the types of Neural Memory Models. Also, give one example of each.

Define Error Correction Learning with examples.