Mechanical Engineering Principles

TORQUE 115

(b) Initial kinetic energy is given by:

I

ω^21

2

=

( 25 )

(

450 × 2 π

60

) 2

2

= 27 .76 kJ

The final kinetic energy is the sum of the initial

kinetic energy and the kinetic energy gained,

i.e. I

ω 22

2

= 27 .76 kJ+ 157 .08 kJ

= 184 .84 kJ.

Hence,

( 25 )ω 22

2

= 184840

from which, ω 2 =

√(

184840 × 2

25

)

= 121 .6 rad/s.

Thus,speed at end of 100 revolutions

=

121. 6 × 60

2 π

rev/min=1161 rev/min

Problem 16. A shaft with its associated

rotating parts has a moment of inertia of

55.4 kg m^2. Determine the uniform torque

required to accelerate the shaft from rest to a

speed of 1650 rev/min while it turns through

12 revolutions.

From above,Tθ=I

(

ω^22 −ω^21

2

)

where angular displacementθ =12 rev= 12 ×

2 π= 24 πrad, final speed,ω 2 =1650 rev/min=

1650

60

× 2 π= 172 .79 rad/s, initial speed,ω 1 =0,

and moment of inertia,I= 55 .4kgm^2.

Hence,torque required,

T=

(

I

θ

)(

ω^22 −ω^21

2

)

=

(

55. 4

24 π

)(

172. 792 − 02

2

)

= 10 .97 kN m

Now try the following exercise

Exercise 46 Further problems on kinetic

energy and moment of inertia

1. A shaft system has a moment of iner-
   tia of 51.4 kg m^2. Determine the torque

required to give it an angular acceleration

of 5.3 rad/s^2. [272.4 N m]

2. A shaft has an angular acceleration of 20
   rad/s^2 and produces an accelerating torque
   of 600 N m. Determine the moment of
   inertia of the shaft. [30 kg m^2 ]


3. A uniform torque of 3.2 kN m is applied
   to a shaft while it turns through 25
   revolutions. Assuming no frictional or
   other resistance’s, calculate the increase
   in kinetic energy of the shaft (i.e. the work
   done). If the shaft is initially at rest and
   its moment of inertia is 24.5 kg m^2 , deter-
   mine its rotational speed, in rev/min, at
   the end of the 25 revolutions.
   [502.65 kJ, 1934 rev/min]


4. An accelerating torque of 30 N m is app-
   lied to a motor, while it turns through
   10 revolutions. Determine the increase in
   kinetic energy. If the moment of inertia
   of the rotor is 15 kg m^2 and its speed
   at the beginning of the 10 revolutions
   is 1200 rev/min, determine its speed at
   the end. [1.885 kJ, 1209.5 rev/min]


5. A shaft with its associated rotating parts
   has a moment of inertia of 48 kg m^2.
   Determine the uniform torque required to
   accelerate the shaft from rest to a speed
   of 1500 rev/min while it turns through 15
   revolutions. [6.283 kN m]


6. A small body, of mass 82 g, is fastened
   to a wheel and rotates in a circular path of
   456 mm diameter. Calculate the increase
   in kinetic energy of the body when the
   speed of the wheel increases from 450
   rev/min to 950 rev/min. [16.36 J]


7. A system consists of three small masses
   rotating at the same speed about the same
   fixed axis. The masses and their radii
   of rotation are: 16 g at 256 mm, 23 g at
   192 mm and 31 g at 176 mm. Determine
   (a) the moment of inertia of the system
   about the given axis, and (b) the kinetic
   energy in the system if the speed of rota-
   tion is 1250 rev/min.

[(a) 2. 857 × 10 −^3 kg m^2 (b) 24.48 J]

8. A shaft with its rotating parts has a
   moment of inertia of 16.42 kg m^2 .Itis
   accelerated from rest by an accelerating