Higher Engineering Mathematics

318 DIFFERENTIAL CALCULUS

Hence, radius of curvature,ρ=

√

√

√

√

[

1 +

(

dy

dx

) 2 ] 3

d^2 y

dx^2

=

√√

√

√

[

1 +

(

1

t

) 2 ]^3

−

1

6 t^3

When t=2, ρ=

√

√

√

√

[

1 +

(

1

2

) 2 ]^3

−

1

6 ( 2 )^3

=

√

( 1. 25 )^3

−

1

48

=− 48

√

( 1. 25 )^3 =−67.08

Now try the following exercise

Exercise 129 Further problems on differen-

tiation of parametric equations

1. A cycloid has parametric equations
   x=2(θ−sinθ), y=2(1−cosθ). Eval-
   uate, atθ= 0 .62 rad, correct to 4 significant

figures, (a)

dy

dx

(b)

d^2 y

dx^2

[(a) 3.122 (b)−14.43]

The equation of the normal drawn to

a curve at point (x 1 ,y 1 ) is given by:

y−y 1 =−

1

dy 1

dx 1

(x−x 1 )

Use this in Problems 2 and 3.

2. Determine the equation of the normal drawn

to the parabolax=

1

4

t^2 ,y=

1

2

tatt=2.

[y=− 2 x+3]

3. Find the equation of the normal drawn to the
   cycloidx=2(θ−sinθ),y=2(1−cosθ)at
   θ=

π

2

rad. [y=−x+π]

4. Determine the value of

d^2 y

dx^2

, correct to 4 sig-

nificant figures, atθ=

π

6

rad for the cardioid

x=5(2θ−cos 2θ),y=5(2 sinθ−sin 2θ).

[0.02975]

5. The radius of curvature,ρ, of part of a sur-
   face when determining the surface tension of
   a liquid is given by:

ρ=

[

1 +

(

dy

dx

) 2 ] 3 / 2

d^2 y

dx^2

Find the radius of curvature (correct to 4 sig-

nificant figures) of the part of the surface

having parametric equations

(a)x= 3 t,y=

3

t

at the pointt=

1

2

(b)x=4 cos^3 t, y=4 sin^3 tatt=

π

6

rad

[(a) 13.14 (b) 5.196]