Question

I was reading David Morin's Introduction to classical mechanics , Chapter Angular Momentum Part I, where I found this paragraph. Invariably, we will calculate angular momentum and torque around either the CM or a fixed point (or a point that moves with constant velocity, but this doesn’t come up often). These are the "safe" origins, in the sense that Eq. (8.46) holds. As long as you always use one of these safe origins, you can simply apply Eq. (8.46) and not worry much about its derivation.

Here Eq. (8.46) is the equation τ=Iα where τ represents torque about the axis, I is the moment of inertia about the axis and α is the angular acceleration. I want to know why is the angular acceleration same about the Centre of mass and the stationary point( Which is the point of contact in pure rolling ). I cannot get the intuition behind this concept. I tried searching in Introduction to Mechanics By Kleppner and Kolenkow but could not find there to.

Answer 1

The angular acceleration is the same around the Centre of Mass and a stationary point in pure rolling because the whole body has the same rate of change of angular velocity. Conservation of angular momentum allows us to apply equations like \(\tau = I\alpha\) consistently when we're calculating around these standard points.

Step-by-step explanation:

The student is asking why angular acceleration (\(\alpha\)) is the same around the Centre of Mass (CM) and a stationary point in pure rolling motion. This is often a point of confusion because the intuition might suggest that different points on a rotating body would have different accelerations.

However, in pure rolling motion, the point of contact with the surface (which we treat as a stationary point) has no linear velocity relative to the surface. Because the entire wheel is moving without slipping, every point on the wheel must have the same angular acceleration. That is, the same rate of change of angular velocity. Therefore, angular acceleration is uniform for the whole body.

Law of conservation of angular momentum implies that in the absence of a net external torque, the angular momentum of a system remains constant. Based on this principle, when calculating angular momentum and torque, we are often interested in the system's moment of inertia (I), which depends on the mass distribution relative to the rotation axis.

For a point mass, the moment of inertia is I = mr\(^2\), where \(m\) is the mass and \(r\) is the perpendicular distance from the rotation axis.

The fact that angular acceleration is consistent at the center of mass and the point of contact during pure rolling allows us to apply equations such as \(\tau = I\alpha\) universally within the system, provided we're referring to one of these 'safe' origins that don't change the system's angular momentum externally.