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Question:

i) A sound wave of frequency 400 Hz travels in air at a speed of 320 $ms^{-1}$ Calculate the phase difference between two points on the wave separated by a distance of 0.2 m along the direction of travel of the wave.

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Question:

a) A Transverse waves propagating on a stretched string encounter another string of different characteristic impedance. (i) Write down the equations of particle displacement due to the incident, reflected and transmitted waves. (ii) Specify the boundary conditions and (iii) use these to obtain expressions for reflection and

transmission amplitude coefficients.

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Question:

Consider N identical masses connected through identical springs of force constant k. The free ends of the coupled system are rigidly fixed at $x=0$ and $x=l.$ The masses are made to execute longitudinal oscillations on a frictionless table.

i) Depict the equilibrium as well as instantaneous configurations.

ii) Write down their equations of motion, decouple them and obtain frequencies of normal modes.

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Question:

d) A body of mass 0.2 kg is suspended from a spring of force constant 80 $Nm^{-1}.$ A

damping force is acting on the system for which γ = 4 Nsm−1. Write down the equation of motion of the system and calculate the period of its oscillations. Now a harmonic force F =10cos10t is applied. Calculate a and θ when the steady state

response is given by a cos(ωt − θ).

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Question:

c) Establish the equation of motion of a damped oscillator. Solve it for a weakly damped oscillator and discuss the significance of the results.

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Question:

b) Consider a particle undergoing simple harmonic motion. The velocity of the particle at position $x_{1}$ is $v_{1}$ and velocity of the particle at position $x^{2}$ is $v_{2}.$ Show that the ratio of time period (T) and amplitude (A) is:

$\frac{T}{A}=2\pi\sqrt{\frac{x_2^{2}-x_{1}^{2}}{v_{2}^{1}x_{2}^{2}-v_{2}^{2}x_{1}^{2}}}$

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Question:

Write the dual of the following LP problem:

Min. $Z=3x_1-2x_2+4x_3$

Subject to

$3x_1+5x_2+4x_3\geq 7$

$6x_1+x_2+3x_3=4$

$7x_1-2x_2-x_3\leq 10$

$x_1,x_2,x_3\leq0$

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Question:

Listed in the table below are the activities and sequencing requirements necessary for the completion of a project.

Activity	Predecessor	Duration in weeks
A	__	6
B	A	24
C	A	6
D	A	12
E	A	9
F	C,D,E	18
G	B,F	12
H	G	24

i) Draw a net work diagram for the project.
ii) Find the critical path and the duration for the completion of the project.

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Question:

i) An object is executing simple harmonic motion. Obtain expressions for its kinetic and potential energies.

ii) A spring mass system is characterized by $k=15 Nm^{-1}$ and $m=0.5kg.$ The system is oscillating with amplitude of 0.40 m. Obtain an expression for the velocity of the block as a function of displacement and calculate its value a $x=0.15 m.$

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Question:

The production department for a company requires 3,600 kg of raw material for manufacturing a particular item per year. It has been estimated that the cost of placing an order is ₹ 36 and the cost of carrying inventory is 25 percent of the investment in the inventories. The price of the raw material is Image ignouassignments-ignouacademy-com--p-ignou-24247 10 per kg.

Determine the following:

i) Economic order quantity
ii) Optimal order cycle time and
iii) Minimum yearly inventory cost.

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Question:

) What should be the radius of a space station spinning with an angular speed of 2 rpm such that an astronaut inside the station experiences artificial gravity with

$g=10ms^-2?$

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Question:

A pendulum bob of mass 50 g is suspended on a string from the ceiling of an elevator which is moving downwards with an acceleration 1.5 $ms^{-2}.$ Draw the free body diagram using the non-inertial frame of reference and determine the tension in the string. (Take $g=10$ $(ms^-2)$

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Question:

An investment company wants to study the investment proposals based on the profit factor. While analyzing a new investment proposal, the company estimated the probability distribution for the profit as follows:

Profit (in thousands)	3	5	7	9	10
Probability	0.1	0.2	0.4	0.2	0.1

Using the random numbers:

19, 7, 90, 2, 57, 28

Simulate the profit of the company for six trials.

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Question:

A satellite going around Earth in an elliptic orbit has a speed of 10 km $s^{-1}$ at the perigee which is at a distance of 227 km from the surface of the earth. Calculate the apogee distance and its speed at that point

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Question:

A proton undergoes a head on elastic collision with a particle of unknown mass which is initially at rest and rebounds with 16/25 of its initial kinetic energy. Calculate the ratio of the unknown mass with respect to the mass of the proton

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Question:

. A proton undergoes a head on elastic collision with a particle of unknown mass which is initially at rest and rebounds with 16/25 of its initial kinetic energy. Calculate the ratio of the unknown mass with respect to the mass of the proton.

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Question:

A solid cylinder of mass 3.0 kg and radius 1.0 m is rotating about its axis with a speed of 40 rad $s^{-1}.$ Calculate the torque which must be applied to bring it to rest in 10s. Wha would be the power required?

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Question:

Solve the following cost minimizing assignment problem.

Image ignouassignments-ignouacademy-com--p-solve-91612

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Question:

The position vector of two particles of mass 4.0 kg and 2.0 kg are, respectively,

$\vec{}r_{1} {}=3t\hat{j}+\hat{j}+2t^{2}\hat{k}$ and $\vec{r}_{2}=3i\hat{}+\left ( t^{2}-1 \right )\hat{j+4t\hat{k}}$ where t is in in metres. Determine the position vector of the centre of mass of the system, the velocity of the cm and the net force acting on the system.