The drawing shows two identical systems of objects; eachconsists of the same thr

Question

The drawing shows two identical systems of objects; eachconsists of the same three small balls connected by massless rods.In both systems the axis is perpendicular to the page, but it islocated at a different place, as shown. The same force of magnitudeF is applied to the same ball in each system (see thedrawing). The masses of the balls are m1 = 9.7kg, m2 = 6.4 kg, and m3 =7.4 kg. The magnitude of the force is F = 486 N.(a) For each of the two systems, determine themoment of inertia about the given axis of rotation.(b) Calculate the torque (magnitude and direction)acting on each system. (c) Both systems start fromrest, and the direction of the force moves with the system andalways points along the 4.00-m rod. What is the angular velocity ofeach system after 5.03 s?

Explanation / Answer

(a)
   the moment of inertia of each system will be
   for the system A
   I = m1 r12+ m2r22 + m3r32
     = (9.70 kg) (0 m)2 + (6.4kg) (3.00 m)2 + (7.40 kg) (5.00 m)2
     = ......... kg.m2
   for the system B
   I = m1 r12+ m2r22 + m3r32
     = (9.7 kg) (5.00 m)2 +(6.4 kg) (4.00 m)2 + (7.4 kg) (0 m)2
     = ......... kg.m2
(b)
   if same force of magnitude F = 486 N the torque willnot be the saeme
   we know that
    = F l
   where l = 3.00 m
   in the system B the lever arm is l = 0,since the line of action of the force passes through the axis
   of rotation
   for the system A the torque will be
    = - F l
      = - (486 N) (3.00 m)
      = ....... N.m
   for the system B the torque will be
    = F l
      = (486 N) (0 m)
      = 0 N.m

(c)
   the equation for the angular velocity is givennby

    = o + t
   but = ( / I)

   where is the net torque and I is themoment of inertia
   in both the systems start from rest the initialangular velocity o = 0 rad/s

   final angular velocity are after t = 10 s
   for system A

     = o + ( / I) t
        = ........ rad/s

   for system B
     = o + ( / I) t
        = ........ rad/s

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