Reactions at point while the system is rotating:
8 kg ball is mounted on a horizontal bar associated to a vertical shaft. Neglecting the mass of the bar and shaft what are the reactions at B and C while the system is rotating at a constant speed of 90 rpm.
Solution
Free body diagram of the system is illustrated in Figure (b) above. Reactions at B, i.e. B_{x} and B_{y} are marked as shown. Likewise, reaction at C is also marked. As the vertical shaft rotates, the ball shall traverse a horizontal circular path of radius 300 mm, inertia force of mrω^{2} shall be acting on the ball as illustrated in the diagram.
Now, we write down dynamic equilibrium equations for the same.
∑ F_{x} = 0, ∴ C_{x } + B_{x} - 8 × 0.3 × ω^{2} = 0
ω = 2 π n/60 =( 2 π × 90)/60 = 9.42 rad / sec
C _{x } + B_{x} = 8 × 0.3 × (9.42)^{2}
∴ C _{x} + B_{x} = 212.97 --------- (1)
∑ F_{y} = 0, B_{y} - 8 × 9.81 = 0 -------- (2)
∴ B_{y} = 78.48 N
∑ M_{ A} = 0, ∴ C_{x} × 0.9 - B_{x} × 0.3 - B_{y} × 0.3 = 0
or, 0.9 C_{x} - 0.3 B_{x} = 23.54
∴ B_{x } = 140 N
C _{x} = 72.9 N
B_{y} = 78.48 N