Problem #1
Determine the moment of inertia of the particle system below to (a) the shaft through the mass of 3m which is perpendicular to the plane? And (b) the shaft connecting mass m and 1.5 m!
The moment of inertia of the particle system is expressed by
I = Σmi.ri2, with m = particle mass and r = distance of the particle to the axis of rotation
(a) for the system above with a shaft through a mass of 3m and perpendicular to the plane, then
Itotal = (2.5 m)(0.2)2 + (4m)(0.2)2 + (m)(0.2√2)2 + (1.5m)(0,2√2)2 + (2m)(0.4)2
Itotal = 0.78m kg.m2
(b) for the system above with a shaft connecting masses m and 1.5 m is
Itotal = (2.5 m)(0.4)2 + (3m)(0.2)2 + (2m)(0.2)2
Itotal = 0.6m kg.m2
Problem #2
For each particle system below. Determine the moment of inertia of the x axis for the system (a) and for the z axis for the system (b).
Answer:
The particle inertia moment is expressed by I = Σmi.ri2, with m = particle mass and r = distance of the particle to the axis of rotation
System (a):
Itotal = (4 kg) (3 m)2 + (2 kg)(- 2 m)2 + (3 kg)(- 4 m)2
Itotal = 92 kg.m2
System (b):
The z axis is an axis perpendicular to the plane so for each object has the same r AO = BO = CO = DO = (22 + 32)1/2 = √13 m, then the moment of total inertia of the system against the z axis is
Itotal, z = (3 kg + 2 kg + 4 kg + 2 kg)(√13 m)2
Itotal, z = 143 kg.m2
Problem #3
Three identical particles are bound to the ends of an ankle right triangle by massless connecting rods. Both sides have the same length a. The moment of inertia of this rigid body for the rotation axis to coincide with the hypotenuse (hypotenuse) of the triangle is. . . .
Answer:
From the picture above it is clear that, two objects that coincide with the rotation axis having an inertial moment of zero, there is only one object that has an AB distance from a shaft which has a moment of inertia of,
Isistem = m (AB)2
With AB2 = a2 - (a√2/2)2 = 3a/4
AB = a√3/2, then
Isistem = m(a√3/2)2 = 3ma2/4
The particle inertia moment is expressed by I = Σmi.ri2, with m = particle mass and r = distance of the particle to the axis of rotation
System (a):
Itotal = (4 kg) (3 m)2 + (2 kg)(- 2 m)2 + (3 kg)(- 4 m)2
Itotal = 92 kg.m2
System (b):
The z axis is an axis perpendicular to the plane so for each object has the same r AO = BO = CO = DO = (22 + 32)1/2 = √13 m, then the moment of total inertia of the system against the z axis is
Itotal, z = (3 kg + 2 kg + 4 kg + 2 kg)(√13 m)2
Itotal, z = 143 kg.m2
Problem #3
Three identical particles are bound to the ends of an ankle right triangle by massless connecting rods. Both sides have the same length a. The moment of inertia of this rigid body for the rotation axis to coincide with the hypotenuse (hypotenuse) of the triangle is. . . .
From the picture above it is clear that, two objects that coincide with the rotation axis having an inertial moment of zero, there is only one object that has an AB distance from a shaft which has a moment of inertia of,
Isistem = m (AB)2
With AB2 = a2 - (a√2/2)2 = 3a/4
AB = a√3/2, then
Isistem = m(a√3/2)2 = 3ma2/4
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