A steel ball with mass m is suspended from the ceiling at the bottom end of a light, 29.0-m-long rope. The ball swings back and forth like a pendulum. When the ball is at its lowest point and the rope is vertical, the tension in the rope is three times the weight of the ball, so T=3mg. Required:
a. What is the speed of the ball as it swings through this point? Express your answer with the appropriate units.
b. What is the speed of the ball if T=mg at this point, where the rope is vertical? Express your answer with the appropriate units.

  a) 23.8 m/s

  b) 0

Explanation:

a) The tension in the rope serves to balance the force of gravity and make the ball deviate from a straight-line path. The latter acceleration is ...

  a = (T -mg)/m = 2g = v²/r . . . . . m is mass

  v = √(2gr) = √(2·9.8 m/s²·29 m) = √(568.4 m²/s²)

  v ≈ 23.8 m/s . . . . approximate speed of the ball at the low point

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b) Using the same analysis, the acceleration required to make the ball deviate from its horizontal path is ...

  a = (T -mg)/m = (mg -mg)/m = 0

and the velocity is ...

  v = √(ar) = 0 . . . . . speed of the ball where the rope is vertical