Mechanical Engineering Principles

150 MECHANICAL ENGINEERING PRINCIPLES

Some typical values of mass and the radius of
gyration are given in Table 13.1, where

A=cross-sectional area

L=length

t=disc thickness

R=radius of the solid disc

R 1 =internal radius

R 2 =external radius

ρ=density

Table 13.1

Component mass k^2

Rod, about mid-point ρAL

L^2

12

Rod, about an end ρAL

L^2

3

Flat disc ρπR^2 t

R^2

2

Annulus ρπ(R^22 −R^21 )t

(R 12 +R^22 )

2

Parallel axis theorem

This is of similar form to the parallel axis theorem
of Chapter 7, where

Ixx=IG+mh^2

Ixx =the mass moment of inertia about thexx
axis which is parallel to an axis passing
through the centre of gravity of the rigid
body, namely atG
IG =the mass moment of inertia of the rigid body
about an axis passing throughGand
parallel to thexxaxis
h =the perpendicular distance between the
above two parallel axes.

Problem 12. Determine the mass moment

of inertia about its centroid for a solid

uniform thickness disc. For the disc, its

radius is 0.2 m, its thickness is 0.05 m, and

its density is 7860 kg/m^3.

From Table 13.1, for a disc, mass,

m=ρπR^2 t

= 7860

kg

m^3

×π×( 0 .2m)^2 × 0 .05 m

= 49 .386 kg

Mass moment of inertia about its centroid,

Io=

mR^2

2

=

49. 386 × 0. 22

2

kg m^2

= 0 .988 kg m^2

Now try the following exercises

Exercise 65 Further problems on rotation

and moment of inertia

1. Calculate the mass moment of inertia of a
   thin rod, of length 0.5 m and mass 0.2 kg,
   about its centroid. [0.004167 kg m^2 ]


2. Calculate the mass moment of inertia of
   the thin rod of Problem 1, about an end.
   [0.01667 kg m^2 ]


3. Calculate the mass moment of inertia of
   a solid disc of uniform thickness about its
   centroid. The diameter of the disc is 0.3 m
   and its thickness is 0.08 m. The density of
   its material of construction is 7860 kg/m^3.
   [0.50 kg m^2 ]


4. If a hole of diameter 0.2 m is drilled
   through the centre of the disc of Problem
   3, what will be its mass moment of inertia
   about its centroid? [0.401 kg m^2 ]

Exercise 66 Short answer questions on

force, mass and acceleration

1. Force is measured in.......


2. The two effects of pushing or pulling an
   object are......or.......


3. A gravitational force gives free-falling
   objects a......in the absence of all
   other forces.


4. State Newton’s first law of motion.


5. Describe what is meant by the inertia of
   an object.