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Chapter 1: Problem 1

Count the number of degrees of freedom in each of the following systems. (a) A small bead sliding on a wire. (b) A lamina moving in its own plane. (c) A double pendulum confined to a vertical plane. This consists of a point mass \(A\) suspended from a fixed point by a thin rod; and a second point mass \(B\) suspended from \(A\) by a second thin rod. The rods are hinged at \(A\). (d) A double pendulum which is not confined to a vertical plane.

Short Answer

Expert verified
(a) a small bead sliding on a wire (b) a lamina moving in its own plane (c) a double pendulum confined to a vertical plane (d) a double pendulum not confined to a vertical plane Answer: (a) 1 degree of freedom (b) 3 degrees of freedom (c) 4 degrees of freedom (d) 4 degrees of freedom

Step by step solution

01

(a) Small bead sliding on a wire

There is only 1 degree of freedom for this system because the bead can only move along the wire. The position of the bead on the wire (length) can be determined by a single coordinate.
02

(b) Lamina moving in its own plane

In this case, the lamina has 3 degrees of freedom. This is because the position of its center of mass requires 2 coordinates to specify its location in the plane (x and y), and an additional coordinate to describe its rotation in the plane (angle).
03

(c) Double pendulum confined to a vertical plane

The double pendulum confined to a vertical plane has 4 degrees of freedom. To determine these, look at each point mass: - For point mass A, it has 1 degree of freedom describing its rotation around the hinge. - For point mass B, it has an additional 1 degree of freedom to describe its rotation around the hinge at A. So, in total, this system has 2 degrees of freedom for each point mass, making it 4 degrees of freedom in total.
04

(d) Double pendulum not confined to a vertical plane

In this case, we must consider the degrees of freedom for each point mass along the three spatial dimensions and their rotations: - For point mass A, it has 2 degrees of freedom to describe its orientation in 3D space (e.g., two angles defining the direction of the thin rod to which it is attached). - For point mass B, it again has 2 degrees of freedom to describe its orientation in 3D space (two angles defining the direction of the second thin rod to which it is attached). Thus, the double pendulum which is not confined to a vertical plane has a total of 4 degrees of freedom.

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