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Inside a fixed hollow cylinder with a vertical axis, a pendulum bob of mass 400 g is executing conical motion with the same axis as the cylinder. The cylinder has radius 30 cm, the string length is 50 cm, and the bob maintains contact with the inner frictionless wall. Which of the following statements are correct? (A) The minimum angular velocity for the bob to maintain wall contact is 5 rad/s (B) The tension in the string is 5 N for all angular velocities (C) At omega = 10 rad/s, the normal force from the wall on the bob is 9 N (D) At omega = 10 rad/s, the tension in the string is 20 N

1. The minimum angular velocity of the bob to maintain wall contact is 5 rad/s
2. The tension in the string is 5 N for all values of angular velocity
3. At omega = 10 rad/s, the bob pushes the wall with a force of 9 N
4. At omega = 10 rad/s, the tension in the string is 20 N

Correct answer: At omega = 10 rad/s, the bob pushes the wall with a force of 9 N

Solution

Since the bob is constrained to r=0.30 m and L=0.50 m, the angle theta is fixed: sin(theta)=0.6, cos(theta)=0.8. Vertical equilibrium gives T=mg/cos(theta)=4/0.8=5 N (constant). Radial: N=m*omega²*r - T*sin(theta) = 0.4*omega²*0.3 - 5*0.6. For N>=0: omega² >= 5*0.6/(0.4*0.3)=25, so omega_min=5 rad/s. At omega=10: N=0.4*100*0.3-3=12-3=9 N.

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