Newton's  laws of motion applied to rigid bodies

Problem:

As shown in the figure, a uniform thin rod of weight W is supported horizontally by two supports, one at each end.  At  t = 0, one of these supports is removed.  Find the force on the remaining support immediately thereafter.

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Solution:

Problem:

A block with mass M hangs from a string that slides over a pulley without friction.  The other end of the string is attached to a massless axel through the center of a hoop of mass M and radius R that can roll without slipping on a flat horizontal surface.  The system is released from rest.  Find the tension in the string.

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Solution:

Problem:

Three cylinders with the same mass m, the same length h, and the same external radius R are initially resting on an inclined plane.  The coefficient of sliding friction on the inclined plane, μ, is known and has the same value for all the cylinders.  The first cylinder is an empty tube with inner radius r and density ρ1, the second is a solid cylinder with density ρ2, and the third has a cavity exactly like the first one, but closed with two negligible mass lids and is filled with a liquid with the same density as the cylinder's walls.  The friction between the liquid and the cylinder wall is considered negligible.  The density of the material of the first cylinder is n times greater than that of the second or of the third cylinder.
(a)  Find the condition for angle θ of the inclined plane with the horizontal, so that none of the cylinders is sliding, i.e. we have pure rolling.
(b)  Find the linear acceleration of the cylinders when we have pure rolling.  Compare these accelerations.
(c)  Find the reciprocal ratios of the angular accelerations of all three cylinders if θ is large and the condition for pure rolling is not satisfied.  Compare these angular accelerations.
(d)  Find the magnitude of the interaction force between the liquid and the walls of the third cylinder when this cylinder is sliding, given that the liquid mass is ml.

Solution:

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?

Solution:

Problem:

Near the surface of the earth a uniform disk of mass M1 and radius R is pivoted on a frictionless horizontal axle through its center.  A small mass M2 is attached to the disk at radius R/2, at the same height as the axle.  This system is released from rest.

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(a)  What is the angular acceleration of the disk immediately after it is released?
(b)  What will be the magnitude of the maximum angular velocity that the disk will reach?

Solution:

Problem:

A dumbbell consists of two spheres A and B, each with volume V, which are connected by a rigid rod.  A has mass M and B has mass 2M.  The distance between the centers of the spheres is d as shown below.  In all parts of this problem assume that the mass and volume of the rod and the moment of inertia of each sphere about its diameter are so small that they can be taken to be zero, and that air resistance can be neglected.

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Solution: