Moment of inertia and CM

Center of mass

Problem:

A uniform carpenter's square has the shape of an L, as shown in the figure.  Locate the center of mass relative to the origin of the coordinate system.

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Solution:


Moment of inertia

Problem:

The four particles in the figure below are connected by rigid rods.

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The origin is at the center of the rectangle.  If the system rotates in the x-y plane about the z-axis with an angular speed of 6 rad/s, calculate
(a) the moment of inertia of the system about the z-axis and
(b) the rotational energy of the system.

Solution:

Problem:

Three particles are connected by rigid rods of negligible mass lying along the y-axis as shown.

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If the system rotates about the x-axis with angular speed of 2 rad/s, find
(a)  the moment of inertia about the x-axis and the total rotational kinetic energy evaluated from ½Iω2, and
(b)  the linear speed of each particle and the total kinetic energy evaluated from ∑½mivi2.

Solution:

Problem:

Two circular metal disks have the same mass M and the same thickness d.  Disk 1 has a uniform density ρ1 which is less than ρ2, the uniform density of disk 2.  Which disk, if either, has the larger moment of inertia about its symmetry axis perpendicular to the plane of the disk?

Solution:

Problem:

Find the fraction of the kinetic energy that is translational and rotational when
(a)  a hoop
(b)  a disc and
(c)  a sphere rolls down an inclined plane of height h.  Find the velocity at the bottom in each case.  Compare with a block sliding without friction down the plane.

Problem:

A uniform rectangular object of mass m with sides a and b (b > a) and negligible thickness rotates with constant angular velocity ω about a diagonal through the center.  Ignore gravity.

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(a)  What are the principal axes and principal moments of inertia?
(b)  What is the angular momentum vector in the body coordinate system?
(c)  What external torque must be applied to keep the object rotating with constant angular velocity about the diagonal?