The instantaneous velocity of point B of the given rod of...

The instantaneous velocity of point $B$ of the given rod of length $0.5$ metre is $3 \mathrm{~m} / \mathrm{s}$ in the represented direction. The angular velocity of the rod for minimum velocity of end $A$ is : (a) $1.5 \mathrm{rad} / \mathrm{s}$ (b) $5.2 \mathrm{rad} / \mathrm{s}$ (c) $2.5 \mathrm{rad} / \mathrm{s}$ (d) none of these

The instantaneous velocity of point $B$ of the given rod of length $0.5$ metre is $3 \mathrm{~m} / \mathrm{s}$ in the represented direction. The angular velocity of the rod for minimum velocity of end $A$ is :
(a) $1.5 \mathrm{rad} / \mathrm{s}$
(b) $5.2 \mathrm{rad} / \mathrm{s}$
(c) $2.5 \mathrm{rad} / \mathrm{s}$
(d) none of these

Key Concepts

Relative Velocity Analysis in Rigid Body Motion

The velocities of two points on a rigid body are interconnected through the relative velocity equation, which states that the velocity of one point equals the velocity of another point plus the additional velocity due to rotation (v_A = v_B + ? × AB). This concept is essential to connect the motion of different points and to set up conditions for extremal (minimum or maximum) velocities.

Optimization of Point Velocities

Given one point’s velocity in a rigid body, it is often of interest to find the conditions under which another point’s velocity is minimized or maximized. This involves geometrically locating the instantaneous center of rotation such that the distance from it to the point in question is optimized, thereby minimizing (or maximizing) the product ?r for that point.

Instantaneous Center of Rotation

Any rigid body undergoing planar motion can be considered as rotating about an instantaneous center of rotation (ICR). This point is special because at that moment every point on the body has a velocity that is perpendicular to the line joining it to the ICR, and the magnitude of the velocity is given by the product of the angular velocity and its distance from the ICR.

Linear and Angular Velocity Relationship

In rigid body kinematics the linear speed of any point on the body is related to the angular speed by the relation v = ?r. This means that the farther the point is from the instantaneous center of rotation, the greater its linear speed; this relationship is fundamental in analyzing the motion of different points on a rotating body.

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