A body is rotating with an angular velocity of 2radians per second about the line joining (1, 2,3) and (2, 3,5) find the velocity of the particle of the body which is momentarily at (3, 5,6)

To find the velocity of a particle in a rotating body, we can use the concept of angular velocity and the position vector of the particle relative to the axis of rotation. In this case, we have a body rotating with an angular velocity of 2 radians per second about the line joining the points (1, 2, 3) and (2, 3, 5). We need to determine the velocity of the particle located at (3, 5, 6).

Step-by-Step Approach

1. Determine the Axis of Rotation

The line of rotation is defined by the two points (1, 2, 3) and (2, 3, 5). We can find the direction vector of this line by subtracting the coordinates of the first point from the second:

• Direction vector = (2 - 1, 3 - 2, 5 - 3) = (1, 1, 2)

2. Find the Position Vector of the Particle

The position vector of the particle at (3, 5, 6) relative to the line of rotation can be calculated. First, we need to find the vector from one of the points on the line (let's use (1, 2, 3)) to the particle:

• Position vector from (1, 2, 3) to (3, 5, 6) = (3 - 1, 5 - 2, 6 - 3) = (2, 3, 3)

3. Calculate the Velocity of the Particle

The velocity of the particle due to the rotation can be found using the formula:

Velocity = Angular Velocity × Position Vector

We need to cross the angular velocity vector with the position vector. The angular velocity vector can be represented as:

• Angular velocity vector = (0, 0, 2) (since it is rotating about the axis defined by the direction vector)

Now, we can compute the cross product:

• Position vector = (2, 3, 3)
• Angular velocity vector = (0, 0, 2)

The cross product is calculated as follows:

• Velocity = (0, 0, 2) × (2, 3, 3)
• Using the determinant method:
• Velocity = (i, j, k)
  | 0 0 2 |
  | 2 3 3 |
  | 0 0 0 |

Calculating this determinant gives:

• Velocity = (0*3 - 2*0, 2*0 - 0*3, 0*3 - 0*2) = (0, 0, 6)

4. Final Result

The velocity of the particle located at (3, 5, 6) is:

• Velocity = (0, 0, 6) m/s

This means that the particle is moving vertically upwards at a speed of 6 m/s due to the rotation of the body. The direction of the velocity vector indicates that it is aligned with the z-axis, which is consistent with the rotation about the defined axis.