IIT BombayDept. of Mechanical EngineeringPh.D. Entrance Exam Sample Question PaperNote:1. All questions are compulsory. There are two parts to the paper. Part A is of 90 minutes duration dealing with Mechanical Engineering topics. Part B deals with General Aptitude part and could be of 90 minutes duration or lesser. There will be 6 questions in Part A, while more questions of the nature given here, will appear in Part B. PartB will have choices but you may be asked to elaborate how you arrived at the correct choice for some questions.2. Make any assumptions and state them in your solution. Use the space below each problem and the back side of the page if necessary.3. Calculators are permitted. In case you do not have a calculator, please simplify to the maximum extent possible.

PART A Mechanical EngineeringProblem 1: The four devices shown in Fig. below rest on frictionless wheels, are restricted to move in the x direction only and are initially held stationary. The pressure at the inlets and outlets of each is atmospheric, and the flow is incompressible. The contents of each device is not known. When released, which devices will move to the right and which to the left ? Explain

Problem 2: The following second order homogeneous differential equation for with constant coefficient is widely encountered in many engineering applications:

Note that is a real number which may be either positive or negative. Determine the general solution of this equation.

Problem 3:A series of cold rolling operations are used to reduce the thickness of a plate from 50 mm down to 25 mm in a reversing two-high mill. Roll diameter = 700 mm and coefficient of friction between rolls and work = 0.15. If the draft is to be equal in each pass, determine the (a) minimum number of passes required, and (b) draft for each pass?

Problem 4:

An open-ended, thin walled cylinder, radius r = 10 cm and thickness t = 1 mm is acted by an internal pressure p and an axial force F. Find the values of p and F if m= 9 MPa, n = 3 MPa and mn= ?. Note that for the case of a thin-walled cylinder under internal pressure p one can assume that the radial stress, r, is zero and the circumferential stress, , is pr/t. Also, in the r--z coordinate system for an axisymmetric problem the three shear stress components are zero.

Problem 5:

For the case of flow of liquid metal through a circular tube, the velocity and temperature profiles at a particular axial location can be approximated as being uniform, and parabolic respectively, where C1 and C2 are constants. Evaluate the bulk mean (mixing cup) temperature.

Problem 6:

The surface roughness can be modeled as a sinusoidal or a square wave. The average surface roughness is given by, Raand root mean square surface roughness is given by, Rq =. Plot the surface profiles and find the ratio of root mean square surface roughness, Rq and average surface roughness, Ra for both cases.

Part B - General Aptitude Test

Questions carrying one markQ1. Choose the word from the options given below that is most nearly opposite inmeaning to the given word: Amalgamate(A) Merge (B) Split (C) Collect (D) Separate

Q2. Which of the following options is the closest in the meaning to the word below:Inexplicable(A) Incomprehensible (B) Indelible(C) Inextricable (D) Infallible

Q3. If then which of the following options isTRUE?(A) P2 = Q3R2 (B) Q2 = PR (C) Q2 = R3P (D) R = P2Q2

Q4. Choose the most appropriate word(s) from the options given below to complete the following sentence.I contemplated________Singapore for my vacation but decided againstit.(A) To visit (B) having to visit (C) visiting (D)for a visit

Q5. Choose the most appropriate word from the options given below to complete the following sentence.If you are trying to make a strong impression on your audience, youcannot do so by being understated, tentative or_____________.

(A) Hyperbolic (B) Restrained(C) Argumentative (D) Indifferent

Questions carrying two marks

Q6. A container originally contains 10 litres of pure spirit. From this container 1 litreof spirit is replaced with 1 litre of water. Subsequently, 1 litre of the mixture isagain replaced with 1 litre of water and this process is repeated one more time.How much spirit is now left in the container?

(A) 7.58 litres (B) 7.84 litres (C) 7 litres (D) 7.29 litres

Q7. Few school curricula include a unit on how to deal with bereavement andgrief, and yet all students at some point in their lives suffer from lossesthrough death and parting.Based on the above passage which topic would not be included in a unit onbereavement?(A) how to write a letter of condolence(B) what emotional stages are passed through in the healing process(C) what the leading causes of death are(D) how to give support to a grieving friend

Q8. P, Q, R and S are four types of dangerous microbes recently found in a human habitat. The area of each circle with its diameter printed in brackets represents the growth of a single microbe surviving human immunity system within 24 hours of entering the body. The danger to human beings varies proportionately with the toxicity, potency and growth attributed to a microbe shown in the figure below

A pharmaceutical company is contemplating the development of a vaccine against the most dangerous microbe. Which microbe should the company target in its first attempt?(A) P (B) Q (C) R (D) S

Q9. A transporter receives the same number of orders each day. Currently, he has some pending orders (backlog) to be shipped. If he uses 7 trucks, then at the end of the 4th day he can clear all the orders. Alternatively, if he uses only 3 trucks, then all the orders are cleared at the end of the 10th day. What is the minimum number of trucks required so that there will be no pending order at the end of the 5th day?(A) 4 (B) 5 (C) 6 (D) 7