unit 1 mm9400 ver 1.1(2014)

This unit is scheduled for E-Learning.

S.I.Units & Dimensions 1 - 1

Statics & Dynamics (MM9400) Version 1.1

1. S.I. UNITS & DIMENSIONS

TIME HEIGHT DEPTH

LENGTH MASS

Objectives: At the end of this unit, students should be able to: Know the common SI Units

Name the basic SI Units. Match the SI Units to the common engineering quantities. Select the appropriate sub-multiple or multiple of the SI Unit.

Understand SI Units

Convert between different metric units. Distinguish between decimal places & significant figured numbers. Write numbers correct to 3 or 4 significant figures. Know the basic dimensions

List the basic dimensions of length, mass and time. State the dimensions of units of Force (e.g. weight), the Newton. Define the Newton. Understand dimensions

Identify the basic dimensions from the units of common physical quantities. Check that an equation is dimensionally homogeneous.


1 - 2 S.I. Units & Dimensions

Version 1.1 Statics & Dynamics (MM9400)

1.1 Physical Quantities Mechanical causes and effects are measured in terms of quantities like distance, velocity, acceleration, time, mass and force. Physical meanings are given by appending units; for instance, 5 metres indicates a certain distance or length, while 5 kilograms refers to the amount of matter possessed by a body. The numeral 5 gives a sense of magnitude or size to the quantity but has no physical meaning by itself; units provide the required physical meaning. Most mechanical quantities comprise only 3 basic independent quantities called dimensions: length, mass and time. 1.2 S.I. Units The Systeme Internationale d' Units (or SI Units) is a widely adopted measurement standard based on the metric system. The basic SI units, i.e. those used for the 3 dimensions, are metre (m), kilogram (kg) and second (s). Angles have no units and are measured in radians. (See Chapter 4). Fig. 1.1 shows some standard multiples and sub-multiples for numerals that are much larger or smaller.

Prefix Symbol Multiplication Factor Giga- G 109

Mega- M 106 kilo- k 103 milli- m 10-3 micro- 10-6

Fig 1.1

Example 1.1 Express the following in SI units with standard multiples or sub-multiples: (a) 2 km (b) 5 mg (c) 1 Mg (d) 36 km/h Solution (a) 2 km = 2 x 103 m (b) 5 mg = 5 x 10-3 x 10-3 kg = 5 x 10-6 kg (c) 1 Mg = 1 x 106 x 10-3 kg = 103 kg or 1 metric tonne (1 t ) (d) 36 km/h = (36 x 103 m) / (60 x 60 s) = 10 m/s




1.3 Significant Figures The accuracy to which a quantity is calculated or measured is implied in the number of significant figures it is rounded off to. For example, 12.3 is a three-significant-figure number which can be any value between 12.25 and 12.34 inclusive; an acceptable error of 0.05 is implied. Writing 12.30 (four significant figures) or even 12.300 (five significant figures) implies a more accurate value; the trailing zeros are significant and their inclusion conveys a greater accuracy. For engineering purposes, quantities are rounded off either to 3 significant figures,

giving relative errors of no more than 1% (i.e. %100100

1 ), or to 4 significant figures, if

expected errors are to be less than 0.1 % (i.e. %1001000

1 ).

Use of decimal places giving absolute errors are unsuitable. For example, an absolute error of 10 mm may be good enough for the height of a tall building but definitely unacceptable for the length of a writing pen! Hence, errors are preferably expressed in percentages, i.e.

Percentage error = 100xValuelTheoretica

Error

The implied accuracy should be realistic. For example, a length of 12.34 mm implies that a ruler has not been used for measurement but a more accurate instrument like the micrometer. Calculated values cannot be of higher accuracy than that of the input values used to compute them. 1.4 Scalar & Vector Quantities Scalar quantities have only magnitude e.g. length (5 m), mass (10 kg), time interval (40 s) and speed (2 m/s). Vector quantities have magnitude and direction e.g. displacement (5 m ), velocity (2 m/s ) and gravitational acceleration (9.81 m/s2 ). Hence, when two vectors are equal, they have the same magnitude and the same direction. A motor car driven at constant speed on the racing track does not have a constant velocity as its direction changes. Force is a vector that changes a body’s state of rest or motion. Its unit is the Newton (N) which is defined as the force required to give a mass of 1 kg an acceleration of 1 m/s2 i.e. 1 N = 1 kg

. 1m/s2. The weight of a body (which is the gravitational force acting on it) is a vector but its mass (which measures only the amount of matter it possesses) is a scalar. Hence, a mass of 1 kg has a weight of 9.81 N .




TUTORIAL: S.I. Units & Significant Figures 1.1 Express the following quantities in SI Units: (a) 0.45 km (b) 80.5 mm (c) 63 g

(d) 0.06 Mg ( 450 m; 0.0805 m; 0.063 kg; 60 kg ) 1.2 Express the following quantities prefixed with standard multiples or sub-

multiples : (a) 12,500 m

(b) 4,600 kg (c) 0.0036 m (d) 85 x 103 kg

( 12.5 km; 4.6 Mg; 3.6 mm; 85 Mg) 1.3 Express the following quantities in SI units to 3 significant figures with standard

multiples or sub-multiples if required: (a) 6 mm2

(b) 426,400 mm3 (c) 0.01356 g/mm3 (d) 90 km/h ( 6.00 x 10-6 m2; 0.426 x 10-3 m3; 13.6 x 103 kg/m3; 25.0 m/s ) 1.4 Round up the following quantities in m2 to 3 significant figures: (a) The area of a heating panel 650 mm long by 330 mm wide

(b) The cross sectional area of a circular rod of 150 mm diameter.

( 0.215 m2; 0.0177 m2) 1.5 Calculate the weight of a 650 kg vehicle, given the acceleration due to gravity is

9.81 m/s2. Explain how your answer conveys the desired level of accuracy. ( 6.38 kN ) 1.6 A cube has edges of 250 1 mm. Calculate the percentage error in:

(a) the length of an edge. (b) the area of a face. (c) the volume.

( 0.4%; 0.8%; 1.2% )




1.5 Dimensional Homogeneity A quantity can only be equal to, added to or subtracted from another when both quantities have the same dimensions, e.g. 1 km = 1 000 m 20 kg + 0.5 t = 520 kg 13 hr 60 min = ½ day However, a quantity may be multiplied or divided by another when both quantities have different dimensions, provided the result has a physical meaning, e.g. 10 m2 x 5 m = 50 m3 (volume) 10 m 5 s = 2 m/s (speed) Hence, an equation must be dimensionally homogeneous, i.e. every individual term to be equated, added or subtracted must have the same dimensions. This enables us to determine the units of a quantity or to check the correctness of an equation or formula. Example 1.2 Find the basic SI units of the following quantities using the given equations. (a) Moment = force x perpendicular distance (see Unit 2) (b) Acceleration = velocity / time (see Unit 4) (c) Force = mass x acceleration (see Unit 5) (d) Momentum = mass x velocity (see Unit 5) Solution [ ] = denotes “has the units” (a) [Moment] = kg m/s2 x m

= kg m2/s2 (b) [Acceleration] = m/s s

= m/s2

(c) [Force] = kg m/s2

(d) [Momentum] = kg x m/s = kg m/s




Example 1.3 The displacement s of a body with initial velocity vo and acceleration a after a time interval of t is given by: s = vo t + ½ at2 . Show that this kinematics equation (see Chapter 4) is dimensionally homogeneous. Solution L.H.S.: [s] = m R.H.S.: [vo t] = m/s x s = m [at2] = m/s2 x s2 = m Example 1.4

Solution Volume of plastic replica vp (n.L)(n.B)(n.H) ……... (n is reduction ratio) Volume of steel structure vs (L)(B)(H)……… (Length x breadth x height)

s

p

m

m

ss

pp

v

v

The plastic replica’s mass, mp =

A 6-m high steel structure weighs 1 tonne.

A 1.5-m high plastic replica is fabricated in the

laboratory. If steel is 10 times heavier than

plastic, the replica weighs ________ kg.

Plastic Replica

1 tonne

1.5

m

6 m

Steel Structure

kg5625.1100010

1

6

5.13




TUTORIAL: Dimensional Homogeneity 1.7 Show that the following equations are dimensionally homogeneous. (a) density = mass / volume (b) pressure = force / area (c) distance = speed x time

(d) force = mass x velocity / time

LHS RHS

(a)

(b)

(c)

(d)

1.8 In a straight-line graph y = mx + c, y has the units m/s and x has the units s.

(a) Find the units of the gradient m and the y-intercept c. (b) Name the quantities represented by each symbol in the equation. (c) What does the area below the line represent?

1.9 A teaching model for a small mechanism enlarges the scale 4 times. The

model’s material density is twice that of the mechanism. The model’s weight is ______ times that of the mechanism.

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unit 1 mm9400 ver 1.1(2014)

Engineering