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Obliquity effect considered.
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Minimum speed $= 240 \ rpm = N_1 , r = 140 \ mm$

Mass of fly ball $= m = 5 \ kg/4 \ kg$

$\therefore$ mean speed = $\frac{N_1 + N_2}{2} = 20(N_2 – N_1)$

$= \frac{240 + N_2}{2} = 20(N_2 – 240)$

$\therefore$ $N_2 = 252.3076 \ rpm$

$N_2 = 212.903 \ rpm$

$\therefore$ $w_1 = \frac{2 \pi \times 240}{60} = 25.132 \ rad/sec$

$\therefore$ $w_2 = \frac{2 \pi \times 252.3076}{60} = 26.4215 \ rad/sec$

$= 22.295 \ rad/sec.$

90 mm = 0.09 m

Radius of rotation @ $N_1 = r_1 = 110 \ mm = 0.110 \ m$

Sleeve lift $= h = 50 \ mm 140 \ mm$

Sleeve arm = horizontal arm $= Y = 100 \ mm 180 \ mm$

Ball arm = vertical arm $= X = 120 \ mm 1100\ mm$

$h = \frac{Y}{X} (r_2 – r_1)$ [$r_2 = 140 \ mm$]

$50 = \frac{100}{120} \ (r_2 – 110)$

$r_2 = 170 \ mm$

$Fc_1 = mr_1 w_1^2$

$= 5 \times 0.11 \times 25.132^2$

$Fc_1 = 347.389N$

$Fc_1 = 157.85 N$

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$a = r – r_1$

$= 140 \ – \ 110 = 30 \ mm$

$a = r – r_1$

$= 150 \ – \ 90 = 25$

{ $X_1 = \sqrt{x^2 – a^2} = \sqrt{120^2 – 30^2} = 116.82 \ mm$

$= \sqrt{100^2 – 25^2} = 96.82$

$Y_1 = \sqrt{y^2 – h_1^2} = \sqrt{ 100^2 – 25^2} = 96.82 \ mm$

$= \sqrt{ 80^2 – 20^2} = 77.4596mm$ }

$\sum M_o = 0\curvearrowright$ +

$\frac{(Mg+S_1)}{2} (Y_1) – Fc_1 (X_1) + mg (a) = 0$

$\frac{S_1}{2} (96.82) – 347.389 (116.82) + 5 \times 9.81 \times 0.03$

$S_1 = 748.29\ N = 369.29\ N$

$b = r_2 – r$

$= 170\ – \ 140 = 30\ mm$

$b = 140\ – \ 115$

$b = 25\ mm$

$X_2 = \sqrt{x^2 – b^2} = \sqrt{120^2 - 30^2} = 116.82\ mm$

$= \sqrt{100^2 – 25^2} = 96.82\ mm$

$Y_2 = \sqrt{y^2 – h^2} = \sqrt{100^2 – 25^2} = 96.82\ mm$

$= \sqrt{80^2 – 20^2} = 77.4596\ mm$

$Fc_2 = mr_2w^2 = 593.3813\ rad/sec$

$fc_2 = 278.357\ rad/sec$

$\sum M_o = 0 \curvearrowright +$

$\frac{(mg + s_2)}{2} (y_2) – Fc_2 (X_2) – mg (b) = 0$

$\frac{s_2}{2} (0.9682) – 593.3812(116.82) – 5 \times 9.81 \times 0.030 = 0$

$S_2 = 1431.880\ N \ 1454.291 \ N$

$721.195\ N$

$s = \frac{S_2 – S_1}{h} = \frac{1454.291 – 748.29}{50}$

$= 14.12\ N/mm \ 8.97$

$\delta = \frac{S_1}{S} = \frac{748.29}{14.121}$

$\delta = 52.98 mm / 42.09$

Spring force @ mid position (S):

$s = \frac{S – S_1}{h_1}$

$14.12 = \frac{S – 748.29}{25}$

$S = 1101.29 \ N / 545.09 \ N$

$N = N mean = \frac{N_1 + N_2}{2} = \frac{240 + 252.3076}{2} = 246.15\ r.p.m.$

$= 20645.5 \ rpm.$

For alteration of speed.

$N^1 = Nmean \sqrt{ \frac{s-f}{S}}$ . . . . . (sleeve moving downward) . . . . . .( S = spring force @ mid-position)

$= 246.15 \sqrt{ \frac{1101.29 – 30}{1101.29}}$

$N” = 2427 rpm / 203.293 rpm.$

$= 246.15 \sqrt{ \frac{1101.29 + 30}{21101.29}}$

$N” = 249.48\ rpm / 208.967$

Alteration in speed $= 249.48\ – \ 242.7$

$= 6.78\ rpm$ . . . . (ANS)

$= 5.674\ rpm.$

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