Question

The CORRECT statement for a rigid body rotating about a fixed axis with angular velocity \(\omega\) is

• A. \(\omega\) is directed perpendicular to the axis of rotation
• B. all the particles move with same speed
• C. \(\omega\) is a scalar quantity
• D. \(\omega\) has no direction
• E. different particles move in different circles

Hint:

For a rigid rotator, \(\omega\) is constant for all points, but \(v\) is not constant. The direction of \(\omega\) is given by the right-hand thumb rule.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This question tests the fundamental properties of rotational motion of a rigid body. A rigid body is one where the distance between any two constituent particles remains constant. When it rotates about a fixed axis, all particles move in circles.
Step 2: Detailed Explanation:
Let's analyze each statement:
(A) \(\omega\) is directed perpendicular to the axis of rotation: This is incorrect. The angular velocity \(\omega\) is an axial vector. Its direction is along the axis of rotation, determined by the right-hand grip rule.
(B) all the particles move with same speed: This is incorrect. All particles of the rigid body have the same angular speed (\(\omega\)). However, their linear speed (v) depends on their perpendicular distance (r) from the axis of rotation, according to the formula \(v = r\omega\). Particles farther from the axis move faster.
(C) \(\omega\) is a scalar quantity: This is incorrect. Angular velocity is a vector quantity, having both magnitude (angular speed) and direction.
(D) \(\omega\) has no direction: This is incorrect, as explained in (A) and (C). It has a well-defined direction along the axis of rotation.
(E) different particles move in different circles: This is correct. Each particle in the rigid body moves in a circular path. The center of each circle lies on the axis of rotation. Particles at different perpendicular distances (r) from the axis will trace out circles of different radii. Only particles at the same perpendicular distance from the axis will move in the same circle.
Step 3: Final Answer:
The correct statement is that different particles move in different circles. This corresponds to option (E).