Mechanical Engineering Principles

130 MECHANICAL ENGINEERING PRINCIPLES

and time,t=10 s. Hence,angular acceleration,

α=

800 × 2 π

60

−

300 × 2 π

60

10

rad/s^2

=

500 × 2 π

60 × 10

= 5 .24 rad/s^2

Problem 4. If the diameter of the shaft in

problem 3 is 50 mm, determine the linear

acceleration of the shaft on its external

surface, correct to 3 significant figures.

From equation (11.11),

a=rα

The shaft radius is

50

2

mm=25 mm= 0 .025 m,

and the angular acceleration,

α= 5 .24 rad/s^2 ,

thus thelinear acceleration,

a=rα= 0. 025 × 5. 24 = 0 .131 m/s^2

Now try the following exercise

Exercise 54 Further problems on linear

and angular acceleration

1. A flywheel rotating with an angular veloc-
   ity of 200 rad/s is uniformly accelerated
   at a rate of 5 rad/s^2 for 15 s. Find the final
   angular velocity of the flywheel both in
   rad/s and revolutions per minute.
   [275 rad/s, 8250/πrev/min]


2. A disc accelerates uniformly from 300
   revolutions per minute to 600 revolutions
   per minute in 25 s. Determine its angular
   acceleration and the linear acceleration of
   a point on the rim of the disc, if the radius
   of the disc is 250 mm.
   [0.4πrad/s^2 ,0.1πm/s^2 ]

11.4 Further equations of motion

From equation (11.3),s = vt, and if the linear

velocity is changing uniformly fromv 1 tov 2 ,then

s=mean linear velocity×time

i.e s=

(

v 1 +v 2

2

)

t ( 11. 12 )

From equation (11.4),θ =ωt, and if the angular

velocity is changing uniformly fromω 1 toω 2 ,then

θ=mean angular velocity×time

i.e θ=

(

ω 1 +ω 2

2

)

t ( 11. 13 )

Two further equations of linear motion may be

derived from equations (11.8) and (11.12):

s=v 1 t+^12 at^2 ( 11. 14 )

and

v 22 =v 12 + 2 as ( 11. 15 )

Two further equations of angular motion may be

derived from equations (11.10) and (11.13):

θ=ω 1 t+^12 αt^2 ( 11. 16 )

and ω 22 =ω^21 +^2 αθ ( 11. 17 )

Table 11.1 summarises the principal equations of

linear and angular motion for uniform changes

in velocities and constant accelerations and also

gives the relationships between linear and angular

quantities.

Problem 5. The speed of a shaft increases

uniformly from 300 rev/min to 800 rev/min

in 10 s. Find the number of revolutions made

by the shaft during the 10 s it is accelerating.