Engineering Mechanics

(^624) „„„„„ A Textbook of Engineering Mechanics
where m 1 , m 2 , m 3 ...... are the masses of the various elements of a body and r 1 , r 2 , r 3 ...... are distances
of the various elements from the fixed line about which the moment of inertia is required to be found
out. In the following pages, we shall discuss the mass moment of inertia of the bodies, which are
important from the subject point of view.
31.7.MASS MOMENT OF INERTIA OF A UNIFORM THIN ROD ABOUT THE
MIDDLE AXIS PERPENDICULAR TO THE LENGTH
Fig. 31.2. Uniform rod.
Consider a uniform thin rod AB of length 2l with O as its mid-point as shown in Fig. 31.2.
Let m= Mass per unit length of the rod.
∴ Total mass of the rod,
M=2ml
Now consider a small strip of length dx at a distance x from the mid-point O. We know that
the mass moment of inertia of the strip about O
=(m dx) x^2 = mx^2 dx ...(i)
The mass moment of inertia of the whole rod may be found out by integrating the above
equation for the whole length of the rod i.e. from – l to + l. Therefore
I=
333
2

• –

() (–)

• 333

l l

l l

xll

mx dx m m

+ +

⎡⎤ ⎡+ ⎤

==⎢⎥ ⎢ ⎥

⎣⎦ ⎣ ⎦

∫

=

2 32

33

ml Ml

= ...(Q M = 2ml)

31.8.MASS MOMENT OF INERTIA OF A UNIFORM THIN ROD ABOUT ONE

OF THE ENDS PERPENDICULAR TO THE LENGTH

Fig. 31.3. Uniform rod.

Consider a uniform thin rod AB of length 2l as shown in Fig. 31.3.

Let m= Mass per unit length of the rod.

∴ Total mass of the rod

M=2 ml

Now consider small strip of length dx at a distance x from one of the ends (say A) as shown in

Fig. 31.3. We know that the mass moment of inertia of the strip about A

=(m dx) x^2 = mx^2 dx ...(i)