Mechanical Engineering Principles

132 MECHANICAL ENGINEERING PRINCIPLES

5 revolutions=5 rev×^2 πrad
1 rev

= 10 πrad

From equation (11.17),ω^22 =ω^21 + 2 αθ

i.e. ( 50 π)^2 =ω^21 +( 2 × 2. 05 × 10 π)

from which, ω^21 =( 50 π)^2 −( 2 × 2. 05 × 10 π)

=( 50 π)^2 − 41 π=24 545

i.e. ω 1 =

√

24 545= 156 .7 rad/s

Thus the initial angular velocity is 156.7 rad/s,
correct to 4 significant figures.

Now try the following exercise

Exercise 55 Further problems on equa-

tions of motion

1. A grinding wheel makes 300 revolutions
   when slowing down uniformly from
   1000 rad/s to 400 rad/s. Find the time for
   this reduction in speed. [2.693 s]


2. Find the angular retardation for the grind-
   ing wheel in question 1. [222.8 rad/s^2 ]


3. A disc accelerates uniformly from
   300 revolutions per minute to 600 revo-
   lutions per minute in 25 s. Calculate the
   number of revolutions the disc makes dur-
   ing this accelerating period.
   [187.5 revolutions]


4. A pulley is accelerated uniformly from
   rest at a rate of 8 rad/s^2. After 20 s the
   acceleration stops and the pulley runs at
   constant speed for 2 min, and then the
   pulley comes uniformly to rest after a
   further 40 s. Calculate:

(a) the angular velocity after the period

of acceleration,

(b) the deceleration,

(c) the total number of revolutions made

by the pulley.

[

(a) 160 rad/s (b) 4 rad/s^2

(c) 12000/πrev

]

11.5 Relative velocity

Quantities used in engineering and science can be

divided into two groups as stated on page 25:

(a) Scalar quantitieshave a size or magnitude

only and need no other information to specify

them. Thus 20 centimetres, 5 seconds, 3 litres

and 4 kilograms are all examples of scalar

quantities.

(b) Vector quantitieshave both a size (or mag-

nitude), and a direction, called the line of

action of the quantity. Thus, a velocity of

30 km/h due west, and an acceleration of

7m/s^2 acting vertically downwards, are both

vector quantities.

A vector quantity is represented by a straight line

lying along the line of action of the quantity, and

having a length that is proportional to the size of

the quantity, as shown in Chapter 3. Thusabin

Figure 11.2 represents a velocity of 20 m/s, whose

line of action is due west. The bold letters,ab,

indicate a vector quantity and the order of the letters

indicate that the lime of action is fromatob.

b a

S

N

W E

0 5 10 15 20 25

Scale : velocity in m/s

Figure 11.2

Consider two aircraftAandBflying at a constant

altitude,Atravelling due north at 200 m/s andB

travelling 30° east of north, writtenN 30 °E,at

300 m/s, as shown in Figure 11.3.

Relative to a fixed point o,oa represents the

velocity ofAandobthe velocity ofB. The velocity

of Brelative toA, that is the velocity at which

Bseems to be travelling to an observer onA,is

given byab, and by measurement is 160 m/s in

a directionE 22 ° N. The velocity ofA relative

toB, that is, the velocity at which A seems to

be travelling to an observer onB,isgivenbyba

and by measurement is 160 m/s in a directionW

22 °S.