In order to understand the parallel-axis theorem and its applications it is given that,
I=Icm+md^2 where Icm is the moment of inertia about an axis passing through the center of mass, m is the total mass of the object, and I is the moment of inertia about another axis, parallel to the one for which Icm is calculated and located a distance d from the center of mass. In this problem you will show that the theorem does indeed work for at least one object: a dumbbell of length 2r made of two small spheres of mass m each connected by a light rod
all axes considered are perpendicular to the plane
Using the definition of moment of inertia, calculate Icm, the moment of inertia about the center of mass, for this object.
The answer to this question is 2mr^2
Using the definition of moment of inertia, calculate Ib(capital i, lower case b), the moment of inertia about an axis through point B, for this object. Point B coincides with (the center of) one of the spheres
expressing the answer in terms of m and r
Now calculate Ib for this object using the parallel-axis theorem.
Express your answer in terms of Icm,m ,r
Using the definition of moment of inertia, calculate Ic, the moment of inertia about an axis through point C, for this object. Point C is located a distance r from the center of mass
Answer expressed in terms of m and r
calculate Ic for this object using the parallel-axis theorem.
Express your answer in terms of Icm,m ,r
I=Icm+md^2 where Icm is the moment of inertia about an axis passing through the center of mass, m is the total mass of the object, and I is the moment of inertia about another axis, parallel to the one for which Icm is calculated and located a distance d from the center of mass. In this problem you will show that the theorem does indeed work for at least one object: a dumbbell of length 2r made of two small spheres of mass m each connected by a light rod
all axes considered are perpendicular to the plane
Using the definition of moment of inertia, calculate Icm, the moment of inertia about the center of mass, for this object.
The answer to this question is 2mr^2
Using the definition of moment of inertia, calculate Ib(capital i, lower case b), the moment of inertia about an axis through point B, for this object. Point B coincides with (the center of) one of the spheres
expressing the answer in terms of m and r
Now calculate Ib for this object using the parallel-axis theorem.
Express your answer in terms of Icm,m ,r
Using the definition of moment of inertia, calculate Ic, the moment of inertia about an axis through point C, for this object. Point C is located a distance r from the center of mass
Answer expressed in terms of m and r
calculate Ic for this object using the parallel-axis theorem.
Express your answer in terms of Icm,m ,r