Sheet (1)
Units and dimensions
1- Using dimensional analysis, put down the dimensions and units in the engineering systems
{pound (Ib), foot (ft), second (s)} and {kilogram (kg), meter (m), second (s)} for the following
engineering quantities:
Density (ρ), specific weight (γ), surface tension (σ), pressure intensity (p), dynamic viscosity
(μ), kinematic viscosity (υ), energy per unit weight, power, liner momentum, angular
momentum, shear stress (τ).
2- Convert the following terms:
[1] γ = 1000 kg/m3 to Ib/ft3
[2] g = 9.81 m/sec2 to ft/sec2
[3] p = 7 kg/cm2 to N/m
[4] γ = 710 dyne/cm 3
to Ib/ft3, N/m3
[5] μ = 4640.84 poise to Ib.sec/ft2, Pa.sec
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[6] 1000 yd = ? in [46] 25 °C = ? °K
[7] 2000 in = ? ft [47] 300 °K = ? °C
[8] 1000 in = ? yd [48] 100 ft/sec = ? mi/hr
[9] 2000 ft = ? in [49] 100 mi / hr = ? ft / sec.
[10] 1000 yd2 = ? in2 [50] 5 gal / min = ? gal/hr
[11] 2000 in2 = ? ft2 [51] 5 pints / min = ? gal / hr
[12] 1000 in2 = ? yd [52] 3 g/cm3 = ? kg/m3
[13] 2000 ft2 = ? in2 [53] 180 kg / m3 = ? g / mm3
[14] 1000 yd3 = ? in3 [54] 200 lb/in2 = ? tons/ft2
[15] 2000 in3 = ? ft3 [55] 2.5 tons / ft2 = ? lb / in2
[16] 1000 in3 = ? yd3 [56] 100 km/hr = ? m/sec
[17] 2000 ft3 = ? in3 [57] 250 m/min = ? km / hr
[18] 1,000,000 mm = ? cm [58] 100 m/sec = ? km/hr
[19] 1,000,000 mg = ? cg [59] 250 km / hr = ? m / min
[20] 1,000,000 mm = ? m [60] 2 cc/sec = ? L/hr
[21] 1,000,000 μg = ? g [61] 2 L / hr = ? cc / min
[22] 1,000,000 mg = ? g [62] 2 m3 = ? L
[23] 1,000,000 cm = ? m [63] 18 L = ? m3
[24] 1,000,000 mL = ? L [64] 4 m2 = ? ft2
[25] 1,000,000 μm = ? cm [65] 4 ft2 = ? m2
[26] 1,000,000 mm = ? km [66] 9 qt = ? m3
[27] 1,000,000 cm = ? km [67] 2 m3 = ? qt
[28] 1,000,000 mg = ? kg [68] 1,000 kg= ? lbs
[29] 1,000,000 g = ? mg [69] 35 lbs = ? kg
[30] 1,000,000 km = ? m [70] 3.2 mg / mm3 = ? oz / in3
[31] 1,000,000 kg = ? g
[32] 1,000,000 km = ? cm
[33] 1,000,000 mL = ? L
[34] 1,000,000 kg = ? mg
[35] 1,000,000 cg = ? g
[36] 50 mi = ? yd
[37] 50 mi = ? in
[38] 1,000,000 tons = ? oz.
[39] 1,000,000 oz = ? tons
[40] 1,000,000 fl. oz. = ? gal
[41] 1,000 fl. oz. = ? quarts
[42] 42 °F = ? °C
[43] 73 °F = ? ° C
[44] 12 °C = ? °F
[45] 70 °C = ? °F
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3- Find the dimensions for the following terms showing which of them is dimensionless:
𝑣. 𝑦 𝜌. 𝑣. 𝑦 𝑣 𝑃 𝐿. 𝑣 2 𝑣 2 𝑃
, , , 2
, , ,
𝜈 𝜇 √𝑔. 𝑦 𝜌. 𝑣 ℎ. 𝑔. 𝑑 𝑔 𝛾
𝑑𝑝 𝜏
𝜌. 𝑣 2 , 𝛾. 𝑦, , , 𝜌. 𝑄. 𝑣, 𝛾. 𝑄. 𝐿
𝑑𝑥 𝑦
𝐶𝐷 ρ 𝑉 2 A
4- The force, F, of the wind blowing against a building is given by F = , where V is the
2
wind speed, ρ the density of the air, A the cross-sectional area of the building, and CD is a
constant termed the drag coefficient. Determine the dimensions of the drag coefficient.
Viscosity
5- The space between a square smooth flat plate (50 x 50) cm2, and a smooth inclined plane
(1:100) is filled with an oil film (S = 0.9) of 0.01 cm thickness. Determine the kinematic
viscosity in stokes if the plate is 2.3 kg. The velocity of the plate = 9 cm/sec.
6- For the shown figure, Calculate the friction force if the plate area is (2m x 3m) and the viscosity
is 0.07 poise.
7- A piston 11.96 cm diameter and 14 cm long works in a cylinder 12 cm diameter. A lubricating
oil which fills the space between them has a viscosity 0.65 poise. Calculate the speed at which
the piston will move through the cylinder when an axial load of 0.86 kg is applied. Neglect the
inertia of the piston.
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8- A piece of pipe 30 cm long weighting 1.5 kg and having internal diameter of 5.125 cm is
slipped over a vertical shaft 5.0 cm in diameter and allowed to fall under its own weight.
Calculate the maximum velocity attained by the felling pipe if a film of oil having viscosity
equals 0.5 Ib.s/ft2 is maintained between the pipe and the shaft.
9- A cylinder of 0.12 m radius rotates concentrically inside of a fixed cylinder of 0.122 m radius.
Both cylinders are 0.30 m long. Determine the viscosity of the liquid which fills the space
between the cylinders if a torque of 1 N.m is required to maintain an angular velocity of 2
rad/s.
10- The thrust of a shaft is taken by a collar bearing provided with a forced lubrication system. The
lubrication system maintains a film of oil of uniform thickness between the surface of the collar
and the bearing. The external and internal diameters of collar are 16 and 12 cm. respectively.
The thickness of oil film is 0.02 cm and coefficient of viscosity is 0.91 poise. Find the horse-
power lost in overcoming friction when the shaft is rotated at a speed of 350 rpm.
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11- The viscosity of liquids can be measured using a rotating cylinder viscometer of the type
illustrated in Figure. In this device the outer cylinder is fixed, and the inner cylinder is rotated
with an angular velocity, ω. The torque, T required to develop ω is measured and the viscosity
is calculated from these two measurements. Develop an equation relating μ, ω, l, Ri and Ro.
Neglect end effects and assume the velocity distribution in the gap is linear.
Surface Tension and capillary effect
12- What is the diameter of a spherical water drop if the inside pressure is 15 N/m2 and the surface
tension is 0.074 N/m.
13- The pressure within a bubble of soapy water of 0.05 cm diameter is 5.75 gm/cm 2 greater than
that of the atmosphere. Calculate the surface tension in the soapy water in S.I. units.
14- Calculate the capillary effect in millimeters in a glass tube of 4 mm diam., when immersed in
(i) water and (ii) in mercury. The temperature of liquid is 20o C and the values of surface
tension of water and mercury at this temperature in contact with air are 0.0075 kg/m and 0.052
kg/m respectively. The contact angle for water = 0 and for mercury = 130°.
15- To what height will water rise in a glass tube if its diameter is (σ =0.072 N/m) a) 1.50 cm and
b) 2.0 mm.
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