# Canadian First Class ME Applied Mechanics

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1.  A conical pendulum has a bob of 2 kg mass attached to the end of a piano wire 1.5 m long.  It is set in notion with a horizontal circular radius of 500 mm.

1) find the angular velocity

2) the time to make one complete revolution of the bob

3) the tension in the wire

Ans: 2.6338 rad/s          Ref Reeds p118

Ans: 2.3856 seconds      Ref JUN89

Ans: 20.8 N                 Ref MAR92

2.  The stabilizing gyroscope of a ship has a mass of 50 tonnes, a radius of gyration of 1.525 m, and rotates about a vertical axis at 900 rev/min.  Calculate:

1) the time required to attain full speed after starting from rest, if the gyro is supplied with a constant power input of 75 kW

2) the torque required to cause precession about the axis in a vertical fore and aft plane at the rate of 1o/s note:

power = Iw(alpha)

torque = Iw(omega) where

I = moment of inertia (kg m2)

w = velocity of spin (rad/sec)

(omega) = velocity of precession

Ans: 3.825 hours         Ref DEC87

Ans: 191.27              Ref JUL89

3. Using strain energy methods, find the mass of a spring required to stop a truck of mass 10 tonnes, travelling at 0.91 m/s.  Assume that there is no compression on the spring before contact with the truck.

note: C(steel) = 7380 kg/m3

G(steel) = 85 GPa

T(steel) = 280 MPa

Ans: 132.4 kg

Ref MAY89
Ref FEB91

4.  An axial thrust of 50 kN is carried by a plain collar type thrust bearing having inner and outer diameters of 250 mm and 400 mm respectively.  Assuming that the coefficient of friction between the thrust surfaces is 0.02, and that the local wear rate of these surfaces is proportional to the pressure and to the rubbing speed, determine the power absorbed in friction of 120 rev/min.

Ans: 2.04 kW

Ref Hannah p209
Ref specimen

5.  During a test on a car, the following information was obtained.

Velocity (m/s)  0  1.2  3.25  6.4  10.2  13.6  15.2  16.0

Time (seconds)  0  2.0  4.0   6.0   8.0  10.0  12.0  14.0

1) Sketch the velocity/time graph

2) find the velocity at 3.5 seconds

3) calculate the distance travelled after 14 seconds

note: graph paper is required for this question

Ans: 4 m/s               Ref MAY89

Ans: 112 m              Ref JUL89

Ref FEB91

6.  A small air compressor is belt driven from a lay shaft in a workshop.  The pulley on the compressor being 300 mm in diameter, and the angle of lap of the belt is 165o.  When the belt is moved from the loose to the fast pulley, it slips for 8 seconds until the compressor attains its constant speed of 300 rev/min.  The flywheel of the compressor has a moment of inertia of 4 km m2 and the friction requires a constant torque of 4 Nm.  If the coefficient of friction is 0.28 during the acceleration period,

1) find the tensions in both reaches of the belt, and also

2) the distance that the belt slips and the energy lost in that time due to the belt slip.

note T1 = euo T2

Ans: 237.4 N             Ref Hannah p220

Ans: 106 N               Ref specimen

Ans: 18.84 m

Ans: 2475.58 J

7.  A belt drive consists of two v-belts in parallel, on grooved pulleys of the same size.  The angle of the groove is 30 degrees.  The cross sectional area of each belt is 750 mm2 and u = 0.12.  The density of the belt material is 1.2 Mg/m3 and the maximum safe stress in the material is 7 MN/m3.

Calculate;

1) the power that can be transmitted between pulleys of 300 mm diameter rotating at 1500 rev/min.

2) the shaft speed in rev/min at which the power transmitted would be at maximum

Ans: 171.66 kW           Ref Hannah p217

Ans: 2807.7 rev/min      Ref JAN88

Ref MAR91

8.  A beam of uniform cross section has a uniformly distributed load of 200 N/m length, it also carries loads as shown.  Draw the shearing force and bending moment diagrams.

PUT DIAGRAM HERE !

Ref FEB84
Ref JUN87

9.  A sluice gate of mass 6 tonnes, is subjected to a normal pressure of 2.5 MN.  It is raised by means of a vertical screw which engages with a screwed bush fixed to the top of the gate.  The screw is rotated by a 37 kW motor running at a maximum speed of 600 rev/min, a bevel pinion on the motor shaft gearing with a bevel wheel of 80 teeth keyed to the vertical screw.  The screw is 125 mm mean diameter and 25 mm pitch.   The u for the screw in the nut is 0.08 and between the gate and its guides is 0.10.  If friction losses, additional to those mentioned above amount to 15% of the total power available, determine the maximum number of teeth for the bevel pinion.

Ans: 14 teeth            Ref Hannah p195

Ref specimen

10.  A gear box is to be arranged for four speeds in approximate geometrical progression, one of which is to be a direct drive.  The driving shaft transmits 30 kW at 2400 rev/min, and the speed of the driven shaft in lowest gear is to be approximately 400 and the layshaft is 180 mm and all teeth are 6 module.  Find:

1) the necessary number of teeth in each pair of gears

2) the torque on the driven shaft

3) the torque on the gear-box frame in lowers gear neglecting friction

Ans: TA + TJ = 24, TB + TK = 36, TC = 48  TD = 12, TE = 41  TF = 19, TH = 27  TG = 33

Ans: 716.22 Nm

Ans: 596.85 Nm in the direction of driving torque

11.  A beam 20 meters long is simply supported at each end. It has a mass of 5000 kg.  It also carries a load of 20 kN at 3 meters from the other end.

1) sketch and dimension the shearing force and bending moment diagrams.

2) calculate the position and magnitude of maximum bending moment

Ans: 10.2039 m from end on 30 kN side

Ans: 247.67 kNm magnitude     Ref FEB91

12.  The pressure on the piston of an engine is 1800 kPa when the crank has turned  through an angle of 30 degrees past top dead centre.  The piston diameter is 220 mm and the ratio of connecting rod length to crank is 4.5.  Determine either graphically or by calculation:

1) the thrust in the connecting rod

2) the force required to overcome the friction of the guide

Ans: 68.883 kN           Ref specimen

Ans: 188.86 N

13.  A curved stationary vane has an inlet angle of 30 degrees and an outlet angle of 20 degrees, both angles being measured from the normal to the vane.  If the vane can withstand a force of 2 kN, determine the maximum diameter of a water jet allowed tangentially on to the inlet of the vane when the supply of water is from a reservoir 20 meters above the vane.  Neglect friction and other losses in the pipe joining the nozzle to the reservoir and at the nozzle.  Take density of water as 1000 kg/m3

Ans: 60 mm               Ref specimen

14.  The distance between the axles of a tanker milk truck is 2.56 m and its center of gravity is 2.45 m above the ground under fully loaded conditions.  Calculate the speed at which the tanker will overturn when traveling around a bend of 50 m radius.  Assume that the road is smooth and flat.

Ans: 57.63 km/hr         Ref Reeds p111

Ref MAY89
Ref FEB91

15.  Three masses A, B, and C of 1.5 kg, 2.0 kg, and 2.5 kg are arranged as illustrated.  The coefficient of friction between B and the table surface is 0.2 and there is negligible friction in the pulleys.  Starting from rest, find the velocity of B after it has traveled 1 meter.

PUT DIAGRAM HERE !

Ans 1.4 m/s              Ref JAN88

Ref JUN89
Ref JUL89

16.  A ship is pitching 10 degrees above and 10 degrees below the horizontal.  Assuming the motion to be simple harmonic, having a period of 12 seconds,  find the maximum angular velocity and angular acceleration of the ship during pitching.

Ans: 0.0914 rad/s        Ref Hannah p79

Ref JUN89

17. 1) State what is meant by the parallel axis theorem.

2) The second moment of area of a rectangle about an axis through its centroid and parallel to the base is given by: BD3 12

Calculate the polar second moment of a square section of side "s"

Ans: s4

Ref: Hannah p86
Ref  DEC87

18. A ball of 2 kg mass traveling at a velocity of 22 m/s overtakes another ball of mass 4.0 kg traveling in the same direction as the first ball but with a velocity of 10 m/s. In elastic conditions prevail and the coefficient of restitution is 0.80, calculate the velocity of the two balls after collision.

Ans:
a) 17. 2 m/s        Ref: Hannah, pg 8

b)  7.6 m/s         Ref  DEC87

19. A torsional pendulum consists of a wire 0.5 m long, 10 mm diameter, find at its upper end and attached at its lower end to a heavy disk having a moment of inertia of 0.06 kg m2.  The modulus of rigidity of the wire is 44 GN/m2. Find the frequency of torsional oscillation of the disk.  If the maximum displacement to one side of the rest position is 5 degrees find the maximum angular velocity and acceleration of the disk.

Ans:
a) 6.039 Hz

Ref Hannah  p78

20. A venturi meter is placed in a horizontal section of a pipeline, 120 mm diameter. The water pressure measured upstream of the meter is 700 kPa and at the throat of the meter, 650 kPa. The throat diameter is 60 mm and the coefficient of discharge is 0.98. Calculate the rate of the flow through the pipe (m3/s).

Ans: 0.0286 m3/s

Ref FEB84
Ref JUN89

21. The impeller of a centrifugal pump has an outside diameter of 1000 mm, and inside diameter of 500 mm, and a vane entrance width of 170 mm. The radial velocity of the water through the impeller is 2.0 m/s when the pump rotates at 300 rev/min Calculate:

1) The width of the impeller vane at exit.

2) The angle of the impeller vane at entrance so that water enters without shock.

Ans:
1) 85 mm           Ref DEC83

2) 14.29o          Ref DEC85

Ref DEC87
Ref JUL89

22.  A canning machine has a particular  component of mass 2.5 kg moving with SHM of amplitude 400 mm and 130 oscillations per minute.  Calculate:

1) The maximum accelerating force upon the component,

2) The accelerating force when the component is displaced 300 mm from mid position.

Ans:
1)185.2 N           Ref FEB84

2)138.9 N           Ref MAY89

Ref FEB91
Ref APR91

23. A tank with sides 2 m long contains 1 m of sea water of density 1.024], with 0.6 m of oil [density 0.8] above. Calculate the force on one side of the tank and the position of the centre of pressure.

Ans:
a) 22.315 kN        Ref: Reeds pg 338

b) 1.086 m  below oil surface.

24. A worm and worm wheel have the following particulars:

Worm:   double threaded worm 70 mm dia., 10 mm pitch,  u = 0.08, collar = 100 mm mean Dia. u = 0.06

Worm wheel: 80 teeth, 80 mm bearings, u = 0.015, output = 15 kW, rev/min = 20

worm square threaded. Determine the input torque on the worm spindle and the overall efficiency.

Ans:
a) 496.8 Nm         Ref Hannah  p192

b) 36%              Ref APR85

Ref DEC86

25. A figure skater has just gone into a pirouette on one skate with both arms and other leg and foot extended. In this position, her angular velocity and moment of inertia are:

[W = 2 pi rad/s]  [I = 4.5 kg/m2]

Neglect friction, calculate her angular when she pulls into the fast spin position. The moment of inertia during fast spin is 1.2 kg m2

Ref MAR91

26. A girder of uniform cross section is 8 m long and supports a informal distributed load of 400 kN. The girder is supported by three columns at the centre and both ends such that all support points are at the same level.

1) calculate the reactions in the columns

2) sketch the bending moment diagrams.

note: the deflections [S1] at midspan due to a uniformly distributed load W and deflection [S2] sue to force P in outer columns are:

s1 = 5WL3         s2 =   PL3     384EI               48EI

Ans:
R1 = R2 = 75 kN     Ref Reeds
R3 = 250 kN            Ref DEC85

27. A steel bar 30 mm diameter and 400 mm long is placed inside a brass tube. Outside diameter 40 mm and inside diameter 32 mm. The tube is 0.125 mm longer than the bar. This compound bar placed between rigid plates arranged so that the axial compressive force of 50 kN can be applied to the assembly. Calculate the stress set up in the tube and bar.  Es = 200 GPa,   Eb = 90 GPa

Ans: os = 40.96 MPa      Ref FEB84

ob = 46.556 MPa     Ref JAN88

28. A machine is driven through a reduction gear of ratio 9:1 by an electric motor. The armature, pinion and shaft  which has a mass moment of inertia of 0.5 kg m2. The mass moment of inertia of the machine components is 40 kg m2. The torque developed by the motor when starting from rest is 30 Nm, the reduction gear efficiency is 95%.  Calculate:

1) the brake power developed by the motor to drive the machine at a constant speed of 160 rev/min.

2) the time required for the speed of the machine to reach 60 rev/min from rest.

Ans:
a) 1.764 kW         Ref Hannah p28

b) 3.15 s           Ref DEC87

Ref MAY89
Ref FEB91

29. A venturi meter having an inlet diameter of 100 mm and a throat diameter of 30 mm is fitted to a pipe line conveying liquid of density 850 kg/m3. A mercury "U" tube is fitted between inlet and throat, and the surfaces of separation have a difference of levels of 200 mm. Assuming a discharge coefficient of 0.96, calculate the mass of flow of the liquid.

note: the density of mercury is 13.6 x 103 kg/m3

Ans: 4.44 kg/s

30. 1) Sketch the controlling force for a Porter type governor using centripetal force for the ordinate and radius for the abscissa.

2) Sketch the controlling force for a Hartwell type governor using a similar coordinate system. a) Illustrate stable, unstable, and isochronous conditions.

3) A Porter type governor has 300 mm arms and the rotating balls each have a mass of 1.8 kg. At the mean speed of 120 rev/min, the arms are at a angle of 30 deg to the vertical axis. Calculate the central dead load of the governor.

Ans: 5.728 kg           Ref Hanna p155

Ref DEC83

31.  A joint is held together by a cotter having a taper of 1:10 equally divided between the two edges. The cotter has a coefficient of friction, u, of 0.18 and is driven into the joint by a force of 500 N. Calculate the force:

1) fastening the joint together.

2) required to eject the cotter.

Ans:
a)1077.7 N          Ref Reeds pg 159
b) 277.6 N          Ref Dec87

32. 1) Derive an expression for the maximum shear stress "T" for a closed-coiled helical spring, where the mean diameter is "D", the wire diameter is "D" and the axial load "W".

2) A close-coiled helical spring has an axle load of 90 N and the mean diameter is 8 times the diameter of the wire Allowing a maximum shear stress of 105 MPa, find the diameter of the wire.

Ans:
1) T = 8WD           Ref JUN89
pi d3
2) 4.18 mm

33. An engine is designed to develop 10 kW of power at a mean speed of 1000 rev/min. Find the moment of inertia in kg m2, of a suitable flywheel, assuming a speed variation of 1.5% of the mean speed and an energy fluctuation equal to 0.9 of the work done per revolution.

Ans: 1.664 kg m2         Ref: Reeds pg 104

34. Water is pumped from a fire engine to the nozzle of a hose being used to fight a fire, the nozzle is 30 m above the pump outlet. The nozzle outlet diameter is 25 mm and the velocity of the water pumped through it is 28 m/s.  Determine the quantity of water flowing through the nozzle per second and the power of the pump if losses are neglected. Take density of water as 1000 kg/m3.

Ans:
a) 13.7445 kg/s
b) 4045 W

35. A Reciprocating engine having a stroke of 600 mm operates at 400 rev/min. calculate the velocity and acceleration of the piston when the crank has turned 30 deg. past top dead center. neglect the effect of connecting rod angularity.

Ans:
a) 6.28 m/s         Ref DEC83
b) 455.86 m/s2      Ref DEC85

36. A spring of stiffness 200 N/m and 0.75 kg mass is set in motion when a mass of 5 kg is attached to the free end. calculate the oscillations:

1) when the spring mass is excluded.

2) when the spring mass is included.

note: 1/3 of spring mass has effect on the hook.

Ans:
1) 1.006 Hz.        Ref Hannah p67
2) 0.982 Hz.        Ref JAN88

37. A uniform ladder rest on a rough horizontal deck leaning against a rough vertical bulkhead. When the ladder makes an angle of 60 deg. to the deck, it is just on the point of slipping. If the coefficient of friction between the foot of the ladder and the deck is 0.27 calculate the coefficient of friction between the top of the ladder and the bulkhead.

Ans: 0.24                Ref Reeds p588

38. A turnbuckle has right hand single start square threads of 8 mm pitch, 28 mm pitch diameter, and the coefficient of friction between screw and nut is 0.15. The device is used to tighten a steel stay having a cross sectional area of 80 mm2, a length of 46 m and a modulus of elasticity, E=140 GPa. If the stay is preloaded to 1.8 kN.  Calculate:

1) energy to tighten the stay one more revolution of the turn buckle.

2) energy lost of friction while tightening the stay.

Ans:
1) 161.1 J          Ref Hannah p190
2) 101.14 J         Ref DEC83

Ref FEB84
Ref MAR85
Ref DEC85
Ref APR91

39. 1) Show that the elastic strain energy of a hollow shaft subject to torsional stress is given by: V = [r2] [D2 + d2] x volume of shaft [4g] [   D2  ]

Where:
u = Elastic strain energy
D = Outside diameter
d = inside diameter
G = Modulus of rigidity

2) A solid shaft 150 mm diameter is to be replaced by a hollow shaft of similar material of same length and have the same mass. The strain energy taken by the hollow shaft is 20% greater torque than can be taken by the solid shaft with the same stress. Find the outside and the inside diameter of the hollow shaft.

Ans:

Ref Reeds #13-13
Ref JUL89

40. A tank 10 m long, 4 m wide, and 6 m high is filled with oil (rd 0.9) and the oil rises to a height of 5 m up a vertical pipe above the top of the tank. Calculate the load on one end plate and on the bottom of the tank.

Ans:
a) 1695.17 kN       Ref Reeds p331
b) 3884.76 kN

41. A bolt and nut falls into a deep tank and the sound of the splash when it strikes the liquid is heard after 2.2 seconds. Estimate the ullage of the tank if the velocity of the sound is 335 m/s a the ambient temperature of the air.

Ans: 22.32 m             Ref specimen

42. The blades of a three blade propeller has masses of 10.20, 10.25 and 10.20 kg . Their centers of gravity are 300 mm, 299 mm and 301 mm respectively from the center of rotation.  the angles are all 120 degrees. Calculate the out of balance force developed at 8000 rev/min.

Ans: 62 N                Ref FEB84

43. A cylindrical pressure vessel as the longitudinal seam welded in the form of a helix. The helical seam is inclined at an angle of 56 degrees to the circumferential seam, the inside diameter of the vessel is 1.8 m, the shell thickness 34 mm, and the working pressure is 3500 kPa. Calculate:

1) the tensile stress normal to the circumferential seam.

2) the tensile stress normal to the oblique seam.

Ans:
a) 52.5 MN/m2       Ref specimen
b) 8.58 MN/m2

44. A satellite is placed in a circular orbit 650 km above the earth's surface at a latitude of 45 degrees. At sea level, g = 9.80665 m/s2, and the earth's mean radius is 6380 km.

1) calculate the speed to maintain the orbit.

2) find the period of one revolution.

Ans:
1) 27127 km/hr      Ref DEC83
2) 1.628 hr            Ref DEC85

Ref FEB90
Ref MAR91

45. A container of mass 2 tonnes is being lowered by a winch at the rate of 0.6 m/s. The cable supporting the load has a cross sectional area of 1200 mm2 and a modulus of elasticity of 200 GPa. When 15 m of cable has been unwound from the which drum, the brake is suddenly applied.

1) find the maximum stress in the cable due to sudden braking.

2) find the maximum extension of the cable.

Ans:
a) 89.44 MN/m2      Ref DEC83
b) 6.71 mm             Ref DEC85

Ref JUN89
Ref JUL89

46. A flywheel has a mass of 100 kg, radius of gyration 0.8m diameter of 0.8 m. It increases speed from 50 rev/min to 2000 rev/min in 10 seconds. Find the accelerating torque.

Ans: 1306.9 Nm           Ref JAN88

47. A spherical gas storage tank is 25 m diameter and constructed of 18 mm plate, the welded seams of which may be considered as being 90% of the plate strength. The yield strength of the material is 245 MPa and a factor of safety of 2.5 is deemed adequate. Calculate the maximum permissible pressure the sphere may safely withstand.

Ans: 254 kPa             Ref DEC87

Ref MAY89
Ref FEB90
Ref FEB91

48. A balloon is rising with a velocity of 5 m/s when a sand bag, used for ballast, is released. At the instant when the sand bag is released, the balloon is 420 m above the earth. Calculate the time required for the sand bag to reach the ground.

Ans: 9.78 seconds        Ref DEC87

49. A cone clutch has a mean surface diameter of 300 mm a face width of 65 mm, and the inclined angle between the friction faces s 30 deg. The clutch lining as a coefficient of friction, u=0.3, and the normal pressure acting upon the surface is 70 pa. Calculate:

1) the maximum power that can be transmitted when rotating at 1200 rev/min.

note: assume uniform wear theory.

Ans: 22.9 kW             Ref Hannah p202

Ref DEC87
Ref DEC89
Ref MAR91

50. The diagram shows a belt drive fitted with a gravity idler. The driver rotates anti-clockwise at 360 rev/min and the coefficient of friction between the belt and pulley is 0.3. Determine the initial belt tension and the power transmitted. Neglect any lag in the belt.

Ans:
a) 175.7 N          Ref Hannah p 222
b) 1987 W          Ref JAN88

Ref JUN89
Ref FEB90

Insert diagram here

51. A mass of 10 kg is laying on a smooth (frictionless) surface of a wheel inclined at 30 degrees.  It is attached by a wire 2 m in length parallel to the plane to centre of the wheel.  The wheel revolves at 10 rev/min.  Find the tension in the wire.

Ans:                  Ref MAY89

Ref JUL89
Ref FEB91

52. A police car equipped with a radar speed detecting device observes a motorist travelling at a speed of 120 km/hr.  The police car starts pursuing the motorist 30 seconds after the initial observation and accelerates to 160 km/hr during a 20 second time interval.  Assume both vehicles maintain their speeds on a straight unimpeded road.

1) sketch a displacement/time graph of the events

2) calculate the time and the distance transverse from the moment of the original observation until the chase ends.

note: graph paper is required for this solution.

Ans: 160 s               Ref FEB84

Ans: 5333.3 m

53.  A cantilever beam is loaded as shown in the diagram below.  Neglecting the weight of the beam, sketch the shear force and bending moment diagrams.

insert diagram here

Ref MAR91

54.  An axial thrust of 50 kN is carried by a plain collar type thrust bearing, having inner and outer diameter of 250 mm, and 400 mm respectively.  Assuming that u between the thrust surfaces is 0.02, and that the local wear rate of these surfaces is proportional to the pressure and to the rubbing speed, determine the power absorbed in friction at a speed of 120 rev/min.

Ans: 2.04 kW             Ref Hannah p209

55.  A bolt and nut falls into a deep tank and the sound of  the splash when it strikes the liquid is heard 2.2 seconds after.  estimate the ullage of the tank if the velocity of sound is 335 m/s at the ambient temperature of the air.

Ans: 22.324 m

56.  A rectangular bar of uniform section 200 mm x 30 mm is used as a structural member and is subjected to a tensile load of 500 kN at 5 mm to one side of the 200 mm axis. Calculate:

1) the maximum and minimum stress induced in the outer fibers of the member

2) sketch the stress distribution across the section

Ans:
95.8 MN/m2 (tensile)
208.3 MN/m2

Ans:
70.8 MN/m2
41.67 MN/m2

57.  A simple pendulum consists of a spherical mass of 10 kg attached to the end of a length of piano wires.  The upper end of the wire attached to a jewelled bearing arrangement on the ceiling to reduce friction.

1) Find the length of wire

2) the angular acceleration 1.5o from the vertical position

Ans: 24.85 m             Ref MAR91