Two points of a rod move with velocities 3vandvperpendicular to the rod and in the same direction, separated by a distancer. Then the angular velocity of the rod is :a)3v/rb)2v/rc)4v/rd)5v/rCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Given:
- The velocities of the two points of the rod are 3v and v perpendicular to the rod and in the same direction.
- The points are separated by a distance r.

To find:
The angular velocity of the rod.

Explanation:
1. Understanding Angular Velocity:
Angular velocity is the rate at which an object rotates or revolves around a fixed axis. It is represented by the symbol ω (omega). Angular velocity is defined as the change in angular displacement (θ) per unit time (t). Mathematically, it can be expressed as:

ω = Δθ/Δt

Where:
ω = Angular velocity
Δθ = Change in angular displacement
Δt = Change in time

2. Relationship between Linear Velocity and Angular Velocity:
The linear velocity of a point on a rotating object is directly related to its angular velocity and the distance from the axis of rotation. The linear velocity (v) can be given by the equation:

v = ωr

Where:
v = Linear velocity
ω = Angular velocity
r = Distance from the axis of rotation

3. Applying the Relationship:
In this problem, we are given the linear velocities of two points on the rod and the distance between them. We can use the relationship between linear velocity and angular velocity to find the angular velocity of the rod.

Let's consider the point with linear velocity 3v. Using the relationship v = ωr, we can write:

3v = ω × r

Similarly, for the point with linear velocity v, we have:

v = ω × r

4. Solving the Equations:
We can solve the above two equations simultaneously to find the value of ω.

From the first equation:
3v = ω × r

Dividing both sides by r:
3v/r = ω

From the second equation:
v = ω × r

Dividing both sides by r:
v/r = ω

5. Final Result:
Comparing the two equations, we can see that the angular velocity ω is the same for both points and is equal to v/r. Therefore, the angular velocity of the rod is 2v/r.

Answer:
The correct answer is option 'B': 2v/r.

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