P01-250 – Relationship Between Angular Speed & Acceleration

Relationship Between Angular Velocity and Angular Acceleration

Effect of angular acceleration on angular velocity:

In this paragraph, we summarize the important results derived from defining instantaneous angular acceleration and instantaneous angular velocity. In particular, we focus on the effect of instantaneous angular acceleration on instantaneous angular velocity.

Consider an object moving along a circular path, counterclockwise, following the different scenarios below.

Scenario 1: angular velocity $\omega $ and angular acceleration $\alpha $ are both positive

If $\omega $ and $\alpha $ are both positive, the particle speeds up in the counterclockwise (positive) direction, as shown by the increasing spacing between consecutive marks on the circular path.

Scenario 2: angular velocity $\omega $ and angular acceleration $\alpha $ are both negative

If $\omega $ and $\alpha $ are both negative, the particle speeds up in the clockwise (negative) direction, as shown by the increasing spacing between consecutive marks on the circular path.

Scenario 3: angular velocity $\omega $ is positive and angular acceleration $\alpha $ is negative

If $\omega $ is positive and $\alpha $ is negative, the particle slows down in the counterclockwise direction as shown by the decreasing spacing between consecutive marks on the circular path.

Note: given enough time, the particle could come to a full stop before speeding up in the clockwise (negative direction) (assuming the angular acceleration continues to act long enough on the particle).

Scenario 4: angular velocity $\omega $ is negative and angular acceleration $\alpha $ is positive

If $\omega $ is negative and $\alpha $ is positive, the particle slows down in the clockwise direction as shown by the decreasing spacing between consecutive marks on the circular path.

Note: given enough time, the particle could come to a full stop before speeding up in the counterclockwise direction (assuming the angular acceleration continues to act long enough on the particle).

Scenario 5: angular acceleration $\alpha $ is zero

If the angular acceleration $\alpha $ is zero, the particle moves at constant angular speed $\omega $ (uniform circular motion).

Conclusion:

The above scenarios illustrate the following conclusion

Acceleration is a quantity that changes the velocity vector, in magnitude or direction, over time.