if you could please show the formulas used, I would really appreciate it! The dr

Question

if you could please show the formulas used, I would really appreciate it!

The drawing shows a systems of objects, which consists of three small balls connected by massless rods. The axis is perpendicular to the page as shown. The force of magnitude F is applied to the m_2 ball (see the drawing). The masses of the balls are m_1 = 9.00 kg, m_2 = 6.00 kg, and m_3 = 7.00 kg. The magnitude of the force is F = 424 N. a. Determine the moment of inertia about the given axis of rotation. b. Calculate the torque magnitude acting on the system. c. The systems start from rest, and the direction of the force moves with the system and always points along the 4.00-m rod. What is the angular velocity of the system after 5.00 s? d. If we change the axis to m_2 ball as shown. Determine the moment of inertia about the given axis of rotation. Calculate the torque magnitude acting on the system.

Explanation / Answer

Given,

Mass, m1 = 9 kg

Mass, m2 = 6 kg

Mass, m3 = 7 kg

Force, F = 424 N

a) the moment of inertia of each system will be,

for the system A, I = m1 r12+ m2 r22 + m3 r32

     = (9 kg) (0 m)2 + (6 kg) (3.00 m)2 + (7 kg) (5.00 m)2 = 229 kg.m2

for the system B, I = m1 r12+ m2 r22 + m3 r32
     = (9 kg) (5.00 m)2 +(6 kg) (4.00 m)2 + (7 kg) (0 m)2 = 321 kg.m2

(b) if same force of magnitude F = 424 N the torque won't be the same. we know that

= F x l (where l = 3.00 m)
   in system B, the lever arm is l = 0, since the line of action of the force passes through the axis of rotation for system A the torque will be
    = - F x l = - (424 N) (3.00 m) = - 1272. N.m

for the system B, the torque will be
    = F x l = (424 N) (0 m) = 0 N.m

(c) Equation for the angular velocity is given by,

    = o + t

but = ( / I) ( where is the net torque and I is the moment of inertia in both the systems start from rest the initial angular velocity o = 0 rad/s)

final angular velocity are after t = 5 s
for system A, = t / I = (-1272 x 5)/ 229 = -27.77 rad/s

   for system B, = t / l = (0 x 5) / 321 = 0 rad/s

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