Consider a mass-spring system with an iron ball (weight 16 pound force) that stretches 8/9 ft with undamped motion. The spring is initially displaced 6 inches upwards from its equilibrium position and given an initial velocity of 1 ft/s downward. Assume the relation mg=hk.

a. Find the displacement at any time t.

b. Find the natural frequency of the mass-spring (i.e. iron ball-spring) system.

c. How many cycles per minutes will the system execute?

d. What would be the amplitude and phase shift?

e. What will its motion be at t= 10 seconds if we pull the ball down from rest by 1.5ft and let it start with zero initial velocity?

d. Now examine two cases with a damping factor of 12 lb/sec and 75 lb/sec with same initial conditions.

Number Theory:

Let p be an odd number. Recall that a primitive root, mod p, is an integer g such that g^p-1 = 1 mod p, and no smaller power of g is congruent to 1 mod p. Some results in this chapter can be proved via the existence of a primitive root(Theorem 6.26)

(c) Given a primitive root g, and an integer a such that a is not congruent to 0 mod p, prove that a is a square modulo p if and only if a = g^e for an even number e. Use this to prove Euler's criterion: a is a square mod p if and only if a^(p-1)/2 = 1 mod p.

i)prove that a cylindrical co-ordinate system is orthogonal.                                     iii) Express the velocity v and acceleration of a particle in cylindrical co-ordinates.                                                   iii) find the square of the element of arc length in cylindrical co-ordinates and determine the corresponding scale factors.                                                           iv) The transformation from retangular to cylindrical co-ordinates is defined by the transformation:.                                   x=pcos#,y=psin#, z=z                                    find the Jacobian of the transformation

Plot π(x), Li(x),x/ln(x) in Mathematica for each of the following ranges: 2≤x≤10,000; 10,000≤x≤20,000; and,100,000≤x≤110,000.

Suppose G is a group and H and H are both subgroups of G.

Let HK={hk, h∈H and k ∈K}

a.give a example such that |HK| not equal to |H| |K|

b. give a example to show f :HK →H ⨯K given by f(hk) = (h,k) may not be well defined.

Hello!

Is someone able to solve the integral of the following function?
f(x) = x sin(pix/2)/(x^4+4).
The boundaries are -inf to +inf.

Thank you!

Prove that the range of a matrix A is equal to the number of singular non-null values of the matrix and Explain how the condition number of a matrix A relates to its singular values.

Let H and K be subgroups of a group G so that for all h in H and k in K there is a k' in K with hk = k'h. Proposition 2.3.2 shows that HK is a group. Show that K is a normal subgroup of HK.

Let a_n denote the number of different ways to color the walls of a five-sided room with n colors if you insist that two walls that meet at a corner must be assigned different colors.

(i) compute a1, a2 and a3 directly

(ii) Find the formula for a_n

Let A = {1,2,3}. Determine all the equivalence relations R on A. For each of these, list all ordered pairs in the relation.

Using induction: Show that player 1 can always win a Nim game in which the number of heaps with an odd number of coins is odd.

Use the branch and bound method to find the optimal solution to the following integer programming problem: maximize 7x1 + 3x2 subject to: 2 x1 + x2 < 9 3 x1 + 2x2 <13 x1, x2 > 0; x1, x2 integer
Instead of using EXCEL Solver to solve this problem directly as an integer programming problem, use EXCEL Solver to solve the LP problems at each branch, with the appropriate constraints added, according to the branch and bound algorithm. Be sure to draw a node and branch diagram to illustrate the procedure, showing the branches that are fathomed, noninteger variables selected, and the optimal solution