Question

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

(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

(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.

(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).

(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.

(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.

Key concepts

Bead on a Wire

When considering a bead sliding on a wire, the concept of degrees of freedom helps us understand how the bead can move. The degrees of freedom refer to the number of independent parameters needed to define the position and movement of an object within a given system.
For a small bead that lies on a wire, its movement is restricted to only one path, which is along the length of the wire. This means that the bead has only one degree of freedom. Essentially, you can define the bead's location using a single coordinate that tells you how far along the wire the bead has traveled. Since the bead can't move sideways or off the wire, its motion is constrained in a very specific way.
This idea of limited movement might seem simple, but it is an excellent example of how constraints can limit the freedom an object has to move.

Lamina Movement

A lamina refers to a thin, flat plate that moves within its own plane, such as a sheet of paper on a desk surface. When we analyze its degrees of freedom, three distinct parameters come into play, allowing us to fully describe its position and orientation.
Firstly, the center of mass of the lamina requires two coordinates \(x, y\) to specify its position within the plane. So even without considering any rotation, the lamina already needs two degrees of freedom merely to locate its center.
Secondly, we must consider orientation or rotational movement. The lamina can rotate around an axis perpendicular to its plane, introducing an angular degree of freedom. Being flat, it doesn't move beyond its plane, and thus its total degrees of freedom remain bound within those three parameters.

• Two coordinates for translation on the plane
• One angle for rotation within the plane

This combination explains why a lamina moving in its own plane enjoys three degrees of freedom.

Double Pendulum

A double pendulum typically consists of two masses joined by rods, free to swing in a vertical plane. The system's complexity and intriguing motion patterns can be explored by understanding its degrees of freedom.
Confined to a vertical plane, the double pendulum exhibits four degrees of freedom, divided between its two masses. Each mass, point A and point B, requires two independent angular coordinates to capture their rotational movement, hence:

• Mass A adds one degree of freedom for its swing relative to the hinge
• Mass B contributes another degree of freedom as it swings from mass A

This dual set of rotations for each mass creates four total degrees of freedom for the vertical plane case.
In contrast, when the pendulum is not confined to a plane, the rods can move through three-dimensional space, heightening complexity. Each mass in this version again needs two angles to define its position but in a 3D context, meaning:

• Mass A has two degrees for its 3D orientation
• Mass B replicates this with its own two degrees

Both configurations maintain four degrees of freedom total, but the state space transitions from a 2D plane to a richer, more dynamic 3D environment.

Three-Dimensional Motion

The concept of three-dimensional motion involves understanding how an object moves in a 3D space with freedom across all axes. This complexity is often mirrored in systems like the double pendulum not confined to a vertical plane.
In 3D motion, unlike the restricted scenarios, every object has the potential to move and rotate independently in three spatial dimensions, leading to a larger degree of freedom. Degrees of freedom in 3D consider both translational and rotational movements across axes:

• Translational freedom requires three coordinates \(x, y, z\) for full spatial representation
• Rotational freedom demands angles for orientation, defining how an object faces or spins in space

Think of a drone flying in an open area. It embodies 3D motion, moving back and forth, side to side, and up and down, while also capable of tilting and rotating. This freedom makes 3D systems particularly challenging to analyze but fascinating due to their breadth of motion possibilities.