Physics · Ch 1 — Rotational Dynamics

Moment of Inertia of a Uniform Ring

1.5.1

Moment of Inertia of a Uniform Ring

Figure 1.13Fig. 1.13: Moment of inertia of a ring — every particle of a uniform ring lies at the same distance R from the central axis, giving I = MR²

Fig. 1.13 — Fig. 1.13: Moment of inertia of a ring — every particle of a uniform ring lies at the same distance R from the central axis, giving I = MR²

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. A thin circular ring of mass M and radius R, drawn as a simple circle (a one-dimensional loop with negligible thickness), rotating about an axis passing through its centre, perpendicular to the plane of the ring (shown as a line or dot-and-cross symbol through the centre). A small representative arc/element of the ring's mass is marked on the circumference, with a radial line of length R drawn from the centre out to that element, emphasising that EVERY point of the ring's mass lies at the same fixed distance R from the axis -- the key geometric fa …

A uniform ring is an object whose mass is spread (practically) uniformly around the circumference of a circle -- a two-dimensional object of negligible thickness, like a bangle or a thin metal hoop. When such a ring of mass M and radius R rotates about its OWN axis (the line through its centre, perpendicular to its plane), every single particle making up the ring's mass lies at EXACTLY the same distance R from that axis -- there is no spread of distances to integrate over at all. The sum $I=\sum m_ir_i^2$ then collapses trivially, since every $r_i=R$ is the same constant: $I=\sum_i m_i R^2=R^2\sum_i m_i=MR^2$ So the moment of inertia of a uniform ring of mass M and radi …

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