applied mechanics

Code No: RR10105 Set No.1

I B.Tech. Regular Examinations, January -2005APPLIED MECHANICS

(Civil Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

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1. (a) Three identical cylinders, each weighing W, are stacked as shown in figure1,on smooth inclined surfaces, each inclined at an angle ‘ θ’ with the horizontal.Determine the smallest angle ‘θ’ to prevent stack from collapsing.

Figure 1:

(b) The boom of a crane is shown in figure2, if the weight of the boom is negligiblecompared with the load W = 60 kN, find the compression in the boom and alsothe limiting value of the tension ‘T’ when the boom approaches the verticalposition.

2. (a) A short semicircular right cylinder of radius ‘r’ and weight ‘w’ rests on ahorizontal surface and is pulled at right angles to its geometric axis by ahorizontal force applied at the middle B of the front edge figure. Find theangle ‘α’ that the flat face will make with the horizontal plane just beforesliding begins if the coefficient of friction at the line of contact A is µ . Thegravity force W must be considered as acting at the center of gravity ‘C’ asshown is the figure3.

(b) The mean diameter of the threads of a square – threaded screw is 50 mm. Thepitch of the thread is 6 mm. The coefficient of friction µ = 0.15. What forcemust be applied at the end of a 600 mm lever, which is perpendicular to thelongitudinal axis of the screw ( i ) to raise a load of 17.5 kN and ( i i ) t o lower the load.

3. (a) Derive an expression for length of an open belt in standard form.

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Figure 2:

Figure 3:

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(b) A belt is running over a pulley of diameter 1200 mm at 200 r.p.m. The angleof contact is 1650 and coefficient of friction between the belt and pulley is 0.3If the maximum tension in the belt is 3000N, find the power transmitted bythe belt.

4. (a) Find the centroid of the plain lamina shown Figure 4

Figure 4:

(b) Find the moment of inertia about the horizontal centroidal axis and about thebase A B{As shown in the Figure 5}

Figure 5:

5. (a) Prove that the mass moment of inertia of a right circular cone of base radiusR and height h, with respect to a diameter of the base is M(3R2 + 2h2)/20where M is the mass of the cone.

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(b) Calculate the mass moment of inertia of a circular cone of base radius 300mmand height 600mm about a line which passes through the mass centre of thecone and which is parallel to the base of the cone. The mass density of thecone is 2500 kg/m3

6. (a) A train is traveling at a speed of 60km/hr. It has to slow down due to certainrepair work on the track. Hence, it moves with a constant retardation of1km/hrper second until its speed is reduced to 15km/hr. It then travels at aconstant speed of for 0.25km/hr and accelerates at 0.5km/hr per second untilits speed once more reaches 60km/hr. Find the delay caused.

(b) The motion of a particle in rectilinear motion is defined by the relations = 2t3− 9t2 + 12t− 10 where s is expressed in metres and t in seconds. Find

i. the acceleration of the particle when the velocity is zero

ii. the position and the total distance traveled when the acceleration is zero.

7. (a) A body weighing 20N is projected up a 200 inclined plane with a velocity of12m/s, coefficient of friction is 0.15. Find

i. The maximum distance S, that the body will move up the inclined plane

ii. Velocity of the body when it returns to it original position.

(b) Find the acceleration of the moving loads as shown in f i gure 6 . Take mass ofP=120kg and that of Q=80Kg and coefficient of friction between surfaces ofcontact is 0.3 .Also find the tension in the connecting string.

Figure 6:

8. Two springs of stiffness k1 and k2 are connected in series. Upper end of the com-pound spring is connected to a ceiling and lower end carries a load ‘W’. Find theequivalent spring stiffness of the system. If the above two springs are connected inparallel then find the equivalent spring stiffness of the system also.

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1. A mast AB supported by a spherical socket at A and horizontal guy wires BC andBD carries a vertical load P at B as shown in Figure7. Find the axial force inducedin each of the three members of this system.

Figure 7:

2. (a) Referring to figure8 the coefficient of the friction is as follows: 0.25 at thefloor, 0.30 at the wall, and 0.20 between blocks. Find the minimum value ofa horizontal force P applied to the lower block that will hold the system inequilibrium.

Figure 8:

(b) Two identical blocks A and B are connected by a rod and rest against verticaland horizontal planes respectively, as shown in figure9. If sliding impendswhen θ = 450, determine the coefficient of friction µ, assuming it to be thesame at both floor and wall.

3. (a) Show that the maximum power can be transmitted at Tmax = 3 Tc

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Figure 9:

(b) A belt embraces the shorter pulley by an angle of 1650 and runs at a speed of1700m/min. Dimensions of the belt are width = 200mm and 8mm thickness.It weight 1000 kg/m3. Determine the maximum power that can be transmittedat the above speed, if the maximum permissible stress in the belt is not toexceed 2.5N/mm2 and µ = 0.25.

4. (a) Define the terms centroid, moment of inertia and radius of gyration.

(b) Compute moment of inertia of hemisphere about its diametral base of radius‘R’.

5. (a) Show that the moment of inertia of a homogenous triangular plate of weightW with respect to its base of width b is Wb2/6g where g is the accelerationdue to gravity.

(b) A right circular cone has the radius of base as 200mm and height 500mm.The mass density of the cone is 7800 kg/m3. Find out the mass moment ofinertia of this cone about a line which passes through the vertex of the coneand which is parallel to the base of the cone.

6. (a) Ram and Rahim are sitting in cars A and B respectively. The cars are 300mapart and at rest. Ram starts the car and moves towards B with an accelera-tion of 0.5m/s2. After three seconds , Rahim starts his car towards A with anacceleration of 1m/s2. Calculate the time and point at which two cars meetwith respect to A.

(b) A projectile is fired at a speed of 800 m/s at an angle of elevation of 500 fromthe horizontal. Neglecting the resistance of air, calculate the distance of thepoint along the inclined surface at which the projectile will strike the inclinedsurface which makes an angle of 150 with the horizontal.

7. (a) A body weighing 20N is projected up a 200 inclined plane with a velocity of12m/s, coefficient of friction is 0.15. Find

i. The maximum distance S, that the body will move up the inclined plane

ii. Velocity of the body when it returns to it original position.

(b) Find the acceleration of the moving loads as shown in figure10. Take mass ofP=120kg and that of Q=80Kg and coefficient of friction between surfaces ofcontact is 0.3 .Also find the tension in the connecting string.

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Figure 10:

8. Determine the frequency of torsional vibrations of the disc shown in (figure11)below, if both the ends of the shaft are fixed and diameter of the shaft is 40mm.The disc has a mass of 600Kg, and a radius of gyration of 0.4m.Taking modulus ofrigidity for the shaft material as 85GN/m2.l1=1m, and l2= 0.8m.

Figure 11:

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1. (a) Define free body diagram, Transmissibility of a force and resultant of a force.

(b) Two identical rollers, each of weight 100 N, are supported by an inclined planeand a vertical wall as shown in Figure12. Assuming smooth surfaces, find thereactions induced at the points of support A, B and C.

Figure 12:

2. The two 50 wedges shown are used to adjust the position of the column under avertical load of 5 kN. Determine the magnitude of the forces P required to raisethe column if the coefficient of friction for all surfaces is 0.40.{As shown in the Figure13}

Figure 13:

3. (a) Derive an expression for length of a crossed belt in standard form.

(b) An engine drives a shaft by means of a belt. The driving pulley of the engineis 3 meters and that in the shaft 2 meters diameter. If the engine runs at150.r.p.m. what will be the speed of the shaft when.

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i. there is no slip

ii. there is a slip of 3% ?

4. (a) Find the centroid of the plain lamina shown Figure14

Figure 14:

(b) Find the moment of inertia about the horizontal centroidal axis and about thebase A B{As shown in the Figure15}

Figure 15:

5. A square prism of cross section 200mm × 200mm and height 400mm stands verti-cally and centrally over a cylinder of diameter 300mm and height 500mm. Calculatethe mass moment of inertia of the composite solids about the vertical axis of sym-metry if the mass density of the material is 2000kg/m3.

6. (a) An airplane is flying horizontally with a velocity of 450 km/hr at an altitudeof 1960 m towards a target on the ground which is to be bombed. Estimate

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where the bomb must be released in order to hit the target and the time oftravel of the bomb. What is the velocity with which the bomb will hit thetarget?. Also find the angle made by the line of sight of the pilot when thebomb is released.

(b) The acceleration of a particle is defined by the relation a =kt-4. Knowing thatv=4m/s when t=2sec and v= 1m/s when t=1 sec, determine the constant ‘k’.Write the equations of motion when x=0 at t=3secs.

7. If Wa:Wb:Wc is in the ratio of 3:2:1 , find the accelerations of the blocks A, B, andC. Assume that the pulleys are weightless.{As shown in the Figure16}

Figure 16:

8. A vertical U-tube manometer contains a liquid of mass density ρ as shown in the(figure17). A sudden increase of pressure on one column forces the level of the liquiddown, When the pressure is released, the liquid column start vibrating. Neglectingthe frictional damping, determine the period of vibration. Comment if the periodis affected by changing the liquid, diameter of the tube or length ‘l’ of the liquidcolumn.

Figure 17:

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1. Calculate the tension T in the guy wire CD and the compression S in each strut ofthe shear-leg derrick shown in Figure18. if the vertical load P = 100 kN.

Figure 18:

2. The three flat blocks are positioned on the 300 incline as shown, and a force Pparallel to the incline is applied to the middle block. The upper block is preventedfrom moving by a wire which attaches it to the fixed support. The coefficient ofstatic friction for each of the three pairs of mating surfaces is shown. Determinethe maximum value which P may have before any slipping takes place.{As shown in the Figure19}

Figure 19:

3. A belt transmits 15 kW from a pulley of 900mm diameter running at 300r.p.m.The angle of lap is 1600 and coefficient of friction is 0.25, thickness of the belt is6mm and its density is 1000kg/m3. Determine minimum width of the belt requiredif stress in belt is limited to 2 N/mm2

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4. (a) Differentiate between centroid and center of gravity.

(b) Determine the product of inertia of shaded area as shown about the indicatex-y axis.{As shown in the Figure20}

Figure 20:

5. (a) Show that the moment of inertia of a thin circular ring of mass M and meanradius R with respect to its geometric axis is MR2.

(b) Find out the mass moment of inertia of a right circular cone of base radius Rand mass M about the axis of the cone.

6. (a) The distance covered by a freely falling body in the last one second of its mo-tion and that covered in the last but one second are in the ratio 5:4. Calculatethe height from which the body was dropped and the velocity with which itstrikes the ground.

(b) A stationary car attains a maximum permissible speed of 80 km/hour in adistance of 40metres. It continues at this speed for a distance of 200 metresand then a uniform retardation brings it to a stop in 10 seconds. How far doesthe car travel from the starting point and what is the total elapsed time?

7. (a) An automobile moving with a uniform velocity of 40Kmph is accelerated byincreasing the traction force by 20%. If the resistance to motion is constant,find the distance traveled before it acquires 50Kmph.Use work-energy method.

(b) A solid cylinder and a sphere are started top of an inclined plane, at the sametime, and both roll without slipping down the plane. If, when the spherereaches the bottom of incline, the cylinder is 12m, what is the total length ’S’of the incline?

8. A spindle of diameter 30mm and of length 3.5m, carries a weight of 280N at one end.The other end of the spindle is fixed. The weight is pulled downwards and releasedso that the spindle is having free longitudinal vibrations. Neglecting the weight ofthe rod, determine the frequency of vibration. Take the modulus of elasticity ofthe material of the spindle as 2×1011N/m2.

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applied mechanics

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