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Asked: 2025-04-05T18:14:33+05:30

How can I determine the direction of angular acceleration from torque and moment arm calculations?

I’m trying to wrap my head around how to determine the direction of angular acceleration from torque and moment arm calculations, but I’m hitting a bit of a wall. I found something on Wikipedia that mentioned angular acceleration is essentially a positive or negative number, depending on whether the angular speed is increasing in a counter-clockwise or clockwise direction. The problem is figuring out how to translate this into actual calculations.

Let’s take an example: I have a torque vector of `<-0.49497476, 0.070710614>` and a moment arm vector of `<-0.5, -3.5>`. Based on the information I found, this should yield a counter-clockwise rotation. However, if I reverse the torque to `<0.49497476, 0.070710614>` and keep the moment arm as `<0.5, -3.5>`, it should indicate a clockwise rotation. But I can’t seem to nail down how to interpret the signs of these vectors in a consistent manner that correlates with the direction of angular acceleration.

In my code, I’m calculating torque based on the angular forces derived from thrusters. I’m looping over each thruster to find the moment arm and applying a smooth force to get an angular force. Then I calculate the torque as the angular force multiplied by the length of the moment arm. So far, using this method, I’m always getting a positive value for the `RotationVelocity`, which feels off because I want to assign the correct sign based on whether it’s spinning clockwise or counter-clockwise.

What I’m missing here? Is there a specific way to adjust the signs in my torque calculations or my moment arm that can give me a clear indication of whether I should be seeing a positive or negative angular acceleration? I know signs can be confusing in physics problems, but I just want to make sure I’m on the right track. How do I ensure the calculations reflect the correct direction of angular acceleration? Any insights on how to approach this would be super helpful!

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    2 Answers

    1. To clearly determine the direction (sign) of angular acceleration from torque and moment arms, especially in a two-dimensional scenario, treat torque as a scalar calculated via the two-dimensional cross product formula: τ = rₓ·F_y - r_y·Fₓ, where r is the moment arm and F is the applied force. This scalar torque value will explicitly give you both magnitude and sign: a positive result indicates counter-clockwise (CCW) angular acceleration, while a negative result indicates clockwise (CW) angular acceleration. For instance, given your vectors (r = <-0.5, -3.5>, F = <-0.49497476, 0.070710614>), the torque computes to a positive scalar, confirming a CCW rotation. Conversely, reversing the torque vector (F = <0.49497476, 0.070710614>) and applying it to (r = <0.5, -3.5>) yields a negative scalar torque, indicating CW rotation.

      In your implementation, explicitly computing torque with the scalar cross product approach ensures consistency and intuitive interpretation of signs. Rather than multiplying torque as a vector by the length of moment arm—which always yields positive magnitude—calculate it as a single scalar value per thruster via the defined cross product formula above. Summing these scalars across all thrusters provides a clear net torque sign, letting you assign correct directionality (positive or negative) to your RotationVelocity and AngularAcceleration. This adjustment will directly solve the confusion you’re facing, providing a clear translation between torque directions in physics and their corresponding signs in your simulation.

    2. Understanding Torque and Angular Acceleration

      It sounds like you’re really diving deep into the physics of angular motion! Let’s break down what you’ve mentioned and see if we can clarify things a bit.

      The relationship between torque (τ), moment arm (r), and angular acceleration (α) is given by the equation: 

      τ = r × F

      Here, F is the force acting at the moment arm r. The direction of the torque vector is crucial, as it indicates the axis of rotation and the direction of rotation (clockwise or counterclockwise).

      Analyzing Your Vectors

      Let’s look at your examples. You have:

      • Torque vector: <-0.49497476, 0.070710614>
      • Moment arm vector: <-0.5, -3.5>

      If this combination yields a counter-clockwise rotation, then the direction of your torque is indeed correct in relation to the moment arm. The torque vector’s negative x component suggests that it’s pushing against the moment arm’s negative y component, causing counter-clockwise motion.

      Reversing the Torque

      When you reverse the torque vector to <0.49497476, 0.070710614>, you’re changing the direction of rotation. If the moment arm remains the same as <0.5, -3.5>, it should indeed result in clockwise rotation.

      Direction of Angular Acceleration

      Now, about your confusion with the signs:

      • Counter-clockwise motion: This is typically considered a positive angular acceleration. If your torque vector points in the direction that encourages this rotation, it should yield a positive α.
      • Clockwise motion: This is considered negative. So if you find that your torque vector is oriented to produce clockwise rotation, you would expect a negative α.

      Wrapping It Up

      Make sure that when you’re calculating your angular velocity (RotationVelocity in your code), you’re checking how the torque direction relates to the moment arm. If you’re always getting a positive value for RotationVelocity, double-check your calculations to see if the signs of the vectors are properly aligned with the expected rotational direction.

      It might be helpful to visualize your vectors! A simple diagram can often make the relationships clearer and help you figure out the signs.

      Keep experimenting with different values, and don’t forget to review the definitions of your vectors and their components. You’re on the right track and getting these concepts down will really help in your programming!