reactions at point while the system is rotating8

Question

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 mrω² 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 × ω² = 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