Mechanical Engineering Principles

184 MECHANICAL ENGINEERING PRINCIPLES

a 2

a 2

a 1

T 2

T 1

F 1 F 2

R 2

R 1

(a) (b)

Figure 16.3

16.2 Motion on a curved banked track

Problem 2. A railway train is required to

travel around a bend of radiusrat a uniform

speed ofv. Determine the amount that the

‘outer’ rail is to be elevated to avoid an

outward centrifugal thrust in these rails, as

shown in Figure 16.4.

L

q

R 1

R 2

Outer

rail

mg

O

h

r CG CF

Figure 16.4

To balance the centrifugal force:

(R 1 +R 2 )sinθ=CF=

mv^2

r

from which, sinθ=

mv^2

r(R 1 +R 2 )

LetR=R 1 +R 2

Then sinθ=

mv^2

rR

( 16. 9 )

Resolving forces vertically gives:

Rcosθ=mg

from which, R=

mg

cosθ

( 16. 10 )

Substituting equation (16.10) into equation (16.9)

gives:

sinθ=

mv^2

rmg

cosθ

Hence tanθ=

v^2

rg

(

since

sinθ

cosθ

=tanθ

)

Thus, the amount that the outer rail has to be

elevated to avoid an outward centrifugal thrust on

these rails,

θ=tan−^1

(

v^2

rg

)

( 16. 11 )

Problem 3. A locomotive travels around a

curve of 700 m radius. If the horizontal

thrust on the outer rail is 1/40thof the

locomotive’s weight, determine the speed of

the locomotive (in km/h). The surface that

the rails are on may be assumed to be

horizontal and the horizontal force on the

inner rail may be assumed to be zero. Takeg

as 9.81 m/s^2.

Centrifugal force on outer rail

=

mg

40

Hence,

mv^2

r

=

mg

40

from which, v^2 =

gr

40

=

9. 81 × 700

40

= 171 .675 m^2 /s^2

i.e. v=

√

171. 675 = 13 .10 m/s

=( 13. 10 × 3. 6 )km/h

i.e.the speed of the locomotive,v= 47 .17 km/h