To be dynamically balanced, ΣM = 0 and ΣF = 0

For $\Sigma$ F_x = 0,

$\Rightarrow$ m₁r₁cos $\theta$ ₁ + m₂r₂cos $\theta$ ₂ + m₃r₃cos $\theta$ ₃ + mrcos $\theta$ = 0

$\Rightarrow$ mr cos $\theta$ = – [m₁r₁cos $\theta$ ₁ + m₂r₂cos $\theta$ ₂ + m₃r₃cos $\theta$ ₃]

$\Rightarrow$ mr cos $\theta$ = – [1 × 100 × cos25° + 2 × 200 × cos 130° + 3 × 300 × cos210°]

$\Rightarrow$ m × 350 × cos $\theta$ = –945.91

$\Rightarrow$ mcos $\theta$ = –2.7 ….(i)

for $\Sigma$ F_y = 0

$\Rightarrow$ m₁r₁sin $\theta$ ₁ + m₂r₂sin $\theta$ ₂ + m₃r₃sin $\theta$ ₃ + mrsin $\theta$ = 0

$\Rightarrow$ mr sin $\theta$ = – [m₁r₁sin $\theta$ ₁ + m₂r₂sin $\theta$ ₂ + m₃r₃sin $\theta$ ₃]

$\Rightarrow$ m × 350 × sin $\theta$ = – [1 × 100 × sin25° + 2 × 200 × sin 130° + 3 × 300 × sin 210°]

$\Rightarrow$ m sin $\theta$ = – 0.289 …(ii)

Squaring and adding equation (i) and (ii) we get,

$\Rightarrow$ m²sin² $\theta$ + m² cos² $\theta$ = (–2.7)² + (–0.289)²

$\Rightarrow$ m²(sin² $\theta$ + cos² $\theta$ ) = 7.37

$\Rightarrow$ m = 2.72 kg

Now, from (ii)

$\theta \approx 186^{\circ}$

Mechanism And Machines