Numerical : 5

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A,B,C and D are four masses carried by a rotating shaft at radii 100 mm, 125 mm, 200 mm and 150 mm respectively. The planes in which the masses revolve are spaced 600 mm apart and the mass of B,C and D are 10 kg, 5 kg and 4 kg respectively. Find the required mass A and the relative angular settings of the four masses so that the shaft shall be in complete balance.

Plane	Mass(m) kg	Radius(r) m	cent.force$\div \omega^2$ (m.r) kg $\cdot$ m	Dist from plane A(L)m	couple $\div \omega^2$ (m.r.l) $kg \ m^2$
A(R.P.)	$m_A$	0.1	0.1 $m_A$	0	0
B	10	0.125	1.25	0.6	0.75
C	5	0.2	1	1.2	1.2
D	4	0.15	0.6	1.8	1.08

Assuming the position of mass B in the horizontal direction OB.
Now the couple polygon is drawn as discussed below :

Draw vector o'b' in the horizontal direction (i.e parallel to OB) and equal to 0.75 $kg \cdot m^2$ to some suitable scale.
From point o' and b', draw vectors o'c' and b'c' equal to 1.2 $kg \cdot m^2$ respectively. These vectors intersects at c'.
Draw OC parallel to vector o'c' and OD parallel to vector b'c'

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In order to find the required mass $A \ (m_A)$ and its angular setting, draw the force polygon to some suitable scale,

enter image description here

$0 \cdot 1 \ m_A \ = \ 0.7 \ \mathrm{kg} \cdot \mathrm{m}^{2}$

$\therefore m_A = 7 \ kg$

Now draw OA parallel to vector do.

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