Is Angular Velocity a Vector?

Angular velocity is indeed a vector quantity, and it is defined as the rate of change of angular displacement or angle with respect to time. It is a property of rigid bodies and therefore is the same regardless of the point of the rigid body under analysis. The direction of the angular velocity vector is perpendicular to the plane of rotation, following the right-hand rule.

Understanding Angular Velocity as a Vector Quantity

Angular velocity, denoted by the symbol ω (omega), is a vector quantity that describes the rate of change of angular displacement of a rigid body. It is measured in radians per second (rad/s) and has both magnitude and direction.

The direction of the angular velocity vector is determined by the right-hand rule. If you point your right thumb in the direction of the rotation, your fingers will curl in the direction of the angular velocity vector.

Relationship between Angular Velocity and Linear Velocity

The relationship between angular velocity and linear velocity is given by the equation:

v = rω

where:
– v is the linear velocity (in m/s)
– r is the distance from the axis of rotation (in m)
– ω is the angular velocity (in rad/s)

This equation shows that the linear velocity is directly proportional to both the distance from the axis of rotation and the angular velocity.

Calculating Velocity Vector from Scalar Angular Velocity and Position Vector

To calculate the velocity vector from a scalar angular velocity and a position vector, you can use the cross product of the angular velocity vector and the position vector. The formula for this is:

v = ω × r

where:
– v is the velocity vector (in m/s)
– ω is the angular velocity vector (in rad/s)
– r is the position vector (in m)

The resulting velocity vector will be perpendicular to the plane of rotation and have a magnitude proportional to the distance from the origin and the angular velocity.

Example:

Suppose a rigid body is rotating around the z-axis with an angular velocity of 2 rad/s. The position vector of a point on the body is r = (2 m, 3 m, 0 m). To find the velocity vector of this point, we can use the cross product:

v = ω × r
  = (2 rad/s) × (2 m, 3 m, 0 m)
  = (0 m/s, 0 m/s, 6 m/s)

The velocity vector of the point is v = (0 m/s, 0 m/s, 6 m/s), which is perpendicular to the plane of rotation (the xy-plane) and has a magnitude of 6 m/s.

Properties of Angular Velocity as a Vector Quantity

1. Direction: The direction of the angular velocity vector is perpendicular to the plane of rotation, following the right-hand rule.
2. Magnitude: The magnitude of the angular velocity vector is the rate of change of the angular displacement with respect to time, measured in radians per second (rad/s).
3. Independence from Point of Analysis: The angular velocity vector is a property of the rigid body and is the same regardless of the point of the rigid body under analysis.
4. Relationship to Linear Velocity: The linear velocity of a point on a rigid body is related to the angular velocity through the equation v = rω, where r is the distance from the axis of rotation.
5. Calculation from Scalar Angular Velocity and Position Vector: The velocity vector can be calculated from the scalar angular velocity and the position vector using the cross product v = ω × r.

Theorems and Formulas Related to Angular Velocity

1. Angular Displacement: The angular displacement θ of a rigid body is the change in the angle of rotation, measured in radians.
2. Angular Velocity: The angular velocity ω is the rate of change of angular displacement with respect to time, measured in radians per second (rad/s).
3. Formula: ω = dθ/dt
4. Linear Velocity: The linear velocity v of a point on a rigid body is related to the angular velocity ω and the distance r from the axis of rotation.
5. Formula: v = rω
6. Velocity Vector from Scalar Angular Velocity and Position Vector: The velocity vector v can be calculated from the scalar angular velocity ω and the position vector r using the cross product.
7. Formula: v = ω × r

Examples and Numerical Problems

1. Example 1: A rigid body is rotating around the z-axis with an angular velocity of 3 rad/s. Find the velocity vector of a point on the body located at the position vector r = (2 m, 3 m, 0 m).

Solution:
v = ω × r
= (3 rad/s) × (2 m, 3 m, 0 m)
= (0 m/s, 0 m/s, 9 m/s)
The velocity vector of the point is v = (0 m/s, 0 m/s, 9 m/s).

1. Numerical Problem 1: A wheel is rotating with an angular velocity of 10 rad/s. Find the linear velocity of a point on the wheel that is 0.5 m from the axis of rotation.

Solution:
v = rω
= (0.5 m) × (10 rad/s)
= 5 m/s
The linear velocity of the point is 5 m/s.

1. Numerical Problem 2: A rigid body is rotating around the z-axis with an angular velocity of 4 rad/s. Find the velocity vector of a point on the body located at the position vector r = (1 m, 2 m, 0 m).

Solution:
v = ω × r
= (4 rad/s) × (1 m, 2 m, 0 m)
= (0 m/s, 0 m/s, 8 m/s)
The velocity vector of the point is v = (0 m/s, 0 m/s, 8 m/s).

Conclusion

In summary, angular velocity is a vector quantity that describes the rate of change of angular displacement of a rigid body. It has both magnitude and direction, with the direction determined by the right-hand rule. The relationship between angular velocity and linear velocity is given by the equation v = rω, and the velocity vector can be calculated from the scalar angular velocity and the position vector using the cross product v = ω × r. Understanding the vector nature of angular velocity is crucial in analyzing the motion of rigid bodies and their associated kinematics.

References

1. Halliday, David, Robert Resnick, and Jearl Walker. Fundamentals of Physics. John Wiley & Sons, 2018.
2. Tipler, Paul A., and Gene Mosca. Physics for Scientists and Engineers. Macmillan, 2018.
3. Serway, Raymond A., Chris Vuille, and Jerry S. Faughn. College Physics. Cengage Learning, 2018.
4. Knight, Randall D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2018.
5. Giancoli, Douglas C. Physics for Scientists and Engineers with Modern Physics. Pearson, 2018.
6. https://physics.stackexchange.com/questions/703506/how-to-calculate-velocity-vector-from-scalar-angular-velocity-and-position-vecto
7. https://www.sciencedirect.com/topics/engineering/angular-velocity-omega
8. https://pressbooks.bccampus.ca/douglasphys1107/chapter/9-1-rotation-angle-and-angular-velocity/
9. https://physics.stackexchange.com/questions/644630/why-is-angular-momentum-a-vector
10. https://quizlet.com/694196119/nardi-final-flash-cards/