Higher Engineering Mathematics, Sixth Edition

The circle and its properties 129

24

22

2

4

y

24

28

23

22 0 2 4 6 x

r^5

4

Figure 13.16

which represents a circle,centre (2,−3)andradius 4,
as stated above.

Now try the following exercise

Exercise 58 Further problemson the

equation of a circle

1. Determine the radius and the co-ordinates of
   the centre of the circle given by the equation
   x^2 +y^2 + 6 x− 2 y− 26 =0.
   [6, (−3, 1)]


2. Sketch the circle given by the equation
   x^2 +y^2 − 6 x+ 4 y− 3 =0.
   [Centre at (3,−2), radius 4]


3. Sketch the curvex^2 +(y− 1 )^2 − 25 =0.
   [Circle, centre (0, 1), radius 5]


4. Sketch the curvex= 6

√[

1 −(y/ 6 )^2

]

[Circle, centre (0, 0), radius 6]

13.6 Linear and angular velocity

Linear velocity

Linear velocityvis defined as the rate of change of
linear displacementswith respect to timet. For motion
in a straight line:

linear velocity=

change of displacement

change of time

i.e. v=

s

t

(1)

The unit of linear velocity is metres per second (m/s).

Angular velocity

The speed of revolution of a wheel or a shaft is usually

measured in revolutions per minute or revolutions per

second but these units do not form part of a coherent

system of units. The basis in SI units is the angle turned

through in one second.

Angular velocity is defined as the rate of change

of angular displacementθ, with respect to time t.

For an object rotating about a fixed axis at a constant

speed:

angular velocity=

angle turned through

time taken

i.e. ω=

θ

t

(2)

The unit of angular velocity is radians per second

(rad/s). An object rotating at a constant speed of

nrevolutions per second subtends an angle of 2πn

radians in one second, i.e., its angular velocityωis

given by:

ω= 2 πnrad/s (3)

From page 124,s=rθ and from equation (2) above,

θ=ωt

hence s=r(ωt)

from which

s

t

=ωr

However, from equation (1)v=

s

t

hence v=ωr (4)

Equation (4) gives the relationship between linear

velocityvand angular velocityω.

Problem 18. A wheel of diameter 540mm is

rotating at

1500

π

rev/min. Calculate the angular

velocity of the wheel and the linear velocity of a

point on the rim of the wheel.

From equation (3), angular velocityω= 2 πnwheren

is the speed of revolution in rev/s. Since in this case