Pure rotation about a fixed axis: When a rigid body moves about a fixed axis (with no translation of the axis), which description best fits its motion?

Introduction / Context:Rigid body kinematics distinguishes between translation, rotation, and general plane motion. Many machine elements (gears, pulleys, flywheels) rotate about a fixed axis, which is a special and very common case in engineering mechanics.

Given Data / Assumptions:

• Axis is fixed in position and orientation in an inertial frame.
• No motion of the axis itself; points on the body trace circular paths about this axis.
• Body is rigid (distances between material points remain constant).

Concept / Approach:

Pure rotation occurs when every point in the body moves in a circle whose centre lies on the fixed axis, with angular position described by a single generalized coordinate (angle). If the axis is fixed, the body has no translational motion of its centroid; translation would imply the axis itself changes position or the body slides bodily, which is not the case.

Step-by-Step Solution:

Verification / Alternative check:

Velocity of a point r from the axis is v = ω × r; acceleration has tangential and normal parts α × r and ω × (ω × r). No net translation of the axis exists.

Why Other Options Are Wrong:

(b) “a circular motion” describes a point, not the body as a whole; (c) translatory motion contradicts a fixed axis; (d) combines both, which is general plane motion (not applicable here); (e) rectilinear motion is incompatible with circles about an axis.

Common Pitfalls:

Describing motion of one point (circular) instead of the entire body (rotary); assuming translation of the centroid when the axis is fixed.

Final Answer:

a rotary motion