Chapter 6 : Centre of Gravity 97
and distance between centre of gravity of the cut out cone and its apex,
33
0.75
4h
yh==We also know that distance between the centre of gravity of the body and apex of the cone (i.e.
54 + 5 = 59 mm),
11 2 2 3 3
12 2vy v y v y
y
vv v+
=
+∴3
3(25 133 45) (16 755 67.5) – 0.1164 0.75
59
25 133 16 755 – 0.1164hh
h×+ × ×
=
+
4
32 261 950 – 0.0873
41 888 – 0.1164h
h=2 471 400 – 6.868 h^3 = 2 261 950 – 0.0873 h^40.0873 h^4 – 6.868 h^3 = – 209 450
Dividing both sides by 0.0873,
h^4 – 78.67 h^3 = –2 399 200 ...(i)
We shall solve this equation by trial and error. First of all, let us substitute h = 10 mm in the left
hand side of equation (i). We find
(10)^4 – 78.67 (10)^3 = – 68 670
We find that answer obtained does not tally with the value of right hand side of equation (i),
and is much less than that. Now let us substitute h = 20 mm in the left hand side of equation (i),
(20)^4 – 78.67 (20)^3 = – 469 360
We again find that the answer obtained does not tally with the right hand side of equation (i),
But it is closer to the value of right hand side than the first case (i.e. when we substituted h = 10 mm.)
Or in other words, the value obtained is still less than the right hand side of equation (i). But the
difference has reduced. Now let us substitute h = 30 mm in the left hand side of equation (i).
(30)^4 – 78.67 (30)^3 = 1 314 100
We again find the answer obtained does not tally with the right hand side of equation (i), But it
is more close to the right hand side than the previous case i.e. when we substituted h = 20 mm. Now
let us substitute h = 40 mm in the left hand side of the equation (i).
(40)^4 – 78.67 (40)^3 = 2474900
Now we find that the answer obtained does not tally with the right hand side of equation (i).
But its value is more than the right hand side of equation (i), In the previous cases, the value of the
answer obtained was less. Thus we find that the value of (h) is less than 40 mm.
A little consideration will show, that as the value of the answer is slightly more than the right
hand side of equation (i). (as compared to the previous answers), the value of (h) is slightly less than
40 mm. Now let us substitude h = 39 mm in the left hand side of the equation (i).
(39)^4 – 78.67 (39)^3 = – 2 153 200
Now we find that the answer obtained is less than the right hand side of equation (i). Thus the
value of (h) is more than 39 mm. Or in other words it is within 39 and 40 mm. This is due to the reason
that when we substitude h = 39 mm, the answer is less and when we substitute h = 40 mm, answer is
more than the right hand side of equation (i), Now let us substitute h = 39.5 mm in the left hand side
of the equation (i).
(39.5)^4 – 78.67 (39.5)^3 = – 2 414 000