How to find instantaneous axis of rotation of a disc rotating about its centre of mass and also having a linear velocity.The disc is moving in space,ie there is no other force acting on it.

To determine the instantaneous axis of rotation of a disc that is both rotating about its center of mass and translating with a linear velocity, we need to consider the motion of the disc in a three-dimensional space. The instantaneous axis of rotation is a line about which the disc is rotating at any given moment. Let's break this down step by step.

Understanding the Motion of the Disc

The disc is undergoing two types of motion simultaneously: rotation and translation. The rotation occurs around its center of mass, while the linear motion is due to its velocity vector. When analyzing this, we can visualize the disc as having a point of rotation that changes depending on its translational velocity.

Identifying the Components

• Rotational Motion: The disc rotates around its center of mass. If we denote the angular velocity as ω, every point on the disc moves in a circular path around this center.
• Translational Motion: The disc has a linear velocity, which we can denote as v. This velocity is directed along a straight line in space.

Finding the Instantaneous Axis of Rotation

The instantaneous axis of rotation can be found using the following approach:

Step 1: Visualize the Motion

Imagine the disc moving through space. The center of mass is moving in a straight line due to the linear velocity, while the points on the edge of the disc are also rotating around the center. The combination of these motions creates a unique axis of rotation.

Step 2: Use the Right-Hand Rule

To determine the direction of the instantaneous axis, apply the right-hand rule. Point your thumb in the direction of the angular velocity vector (which is perpendicular to the plane of the disc) and curl your fingers in the direction of rotation. This will give you the orientation of the instantaneous axis.

Step 3: Determine the Instantaneous Axis Location

The instantaneous axis of rotation will not only be oriented according to the right-hand rule but will also be located at a point that is a function of both the rotation and translation. Specifically, the instantaneous axis will be a line that passes through the center of mass and extends in the direction determined by the combination of the linear velocity and the angular velocity.

Mathematical Representation

Mathematically, if we denote the position vector of the center of mass as r and the linear velocity as v, the instantaneous axis can be represented as:

v = ω × r

Here, ω is the angular velocity vector, and r is the radius vector from the center of mass to any point on the disc. The cross product gives a vector that is perpendicular to both v and r, indicating the direction of the instantaneous axis of rotation.

Example for Clarity

Consider a disc with a radius of 1 meter rotating at an angular velocity of 2 radians per second while moving with a linear velocity of 3 meters per second. The center of mass moves in a straight line, while points on the edge of the disc rotate around the center. The instantaneous axis of rotation will be a line that passes through the center of mass and is oriented according to the right-hand rule, taking into account both the rotation and the linear motion.

In summary, the instantaneous axis of rotation for a disc that is both rotating and translating can be found by visualizing the motion, applying the right-hand rule, and using the relationship between linear and angular velocities. This understanding is crucial in fields such as physics and engineering, where the dynamics of rotating bodies are frequently analyzed.