Sign conventions in torque and non-uniform circular motion

In summary, sign conventions in torque and non-uniform circular motion are essential for understanding the direction and magnitude of forces and rotational effects. Torque is often defined with a right-hand rule, where counterclockwise rotation is positive and clockwise is negative. In non-uniform circular motion, the acceleration can be broken down into tangential and centripetal components, with sign conventions indicating their directions; tangential acceleration aligns with the direction of motion while centripetal acceleration always points inward towards the center of the circular path. Consistent use of these conventions ensures accurate analysis and calculations in physics.
  • #1
DrBanana
51
4
Sorry for the overly general title but my problem is regarding a specific problem: find the net force on the bob of a pendulum as a function of , the angle it makes with the vertical (assuming the observer is stationary with respect to point from which the string is hung and the vertical beam of the pendulum).

The easy way: If the maximum angular displacement is , then the tension for a general angle will be . This force points along the string, so we know its direction. The only other force is the weight. To solve the problem, we just take the resultant of these two forces.

The hard way: I'll abbreviate a 'Leap of faith' as LOF. The bob undergoes non-uniform circular motion (behold my epic drawing skills).
Screenshot_20240503_182132.png
It's position can be described with a vector such that . By definition, velocity is the derivative of the position: . This is where I first run into problems. If I were measuring with its starting position on the positive x axis, I could just define (the angular speed) as the derivative of with respect to time. However, can I do that here? I must make a choice, and the particular choice I make here is that I let the displacement of the bob along the circular arc (starting from the x axis) take negative values for clockwise directions, so that the displacement . Then we can see that (LOF number 1). If we differentiate the velocity vector again, we get , where is the acceleration vector and is the angular acceleration, which, by definition, is the derivative of the angular speed. It just so happens that .

We know that , where is torque, is the moment of inertia and alpha is the angular acceleration. LOF number two: the angular acceleration formulated in the previous paragraph, is the same as this angular acceleration.

We also know that . Then so for small values of . LOF number 3 is, we can actually write this equation. This is perhaps my main point of confusion. For springs I can understand why the restoring force (and thus the acceleration) should have a sign opposite to the displacement, and for pendulums also I can sort of get a feel for why the torque needs to be 'opposite' in sign to the angular displacement, although not fully. For the spring example, the force vector and the displacement vector are in the same plane, so there is a meaning of the word opposite. However torque as a vector extends out from the page, and there is no vector quantity on the right hand side of the equation that does that. I thought that with these mechanics problems you can either write equations of vectors, or of their magnitudes (no directions), but not something in between.

Anyway, assuming I could write that, it would be possible to solve the differential equation and find the term (we would need to specify the maximum angular displacement as well) in terms of . After that we could plug in any value of theta in the net acceleration equation and find the corresponding force.

Using both methods, I find that the results are approximately equal. Are all of the leaps of faith I took luck, or did those work, and if so, why?
 
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  • #2
DrBanana said:
If I were measuring with its starting position on the positive x axis, I could just define (the angular speed) as the derivative of with respect to time. However, can I do that here?
Why not? Let be the angle w.r.t. the positive x-axis increasing clockwise and be as shown in your drawing. Then

This says that one increases at the same rate as the other one decreases.
 

FAQ: Sign conventions in torque and non-uniform circular motion

What is the sign convention for torque?

The sign convention for torque typically follows the right-hand rule. If you curl the fingers of your right hand in the direction of the force applied, your thumb points in the direction of the torque. Torque is considered positive if it causes a counterclockwise rotation and negative if it causes a clockwise rotation.

How do we determine the direction of torque in a system?

The direction of torque can be determined by the cross product of the position vector (from the pivot point to the point of force application) and the force vector. The resulting direction, as given by the right-hand rule, indicates whether the torque is positive or negative based on its rotational effect.

What is the significance of non-uniform circular motion?

Non-uniform circular motion refers to the motion of an object traveling in a circular path with a variable speed. This type of motion is significant because it involves both tangential acceleration (due to changes in speed) and centripetal acceleration (due to changes in direction), affecting the net forces acting on the object and its overall dynamics.

How do we apply sign conventions in non-uniform circular motion?

In non-uniform circular motion, the tangential acceleration is considered positive if it is in the direction of the velocity vector (speeding up) and negative if it opposes it (slowing down). The centripetal acceleration is always directed towards the center of the circular path and is treated as positive, while the net force acting on the object is determined by the vector sum of the tangential and centripetal components.

What are common mistakes when applying sign conventions in torque and non-uniform circular motion?

Common mistakes include misidentifying the direction of torque based on the force application, confusing the signs of tangential and centripetal accelerations, and neglecting to consider the effects of both types of acceleration in the analysis of forces. It's crucial to consistently apply the right-hand rule and carefully analyze the components of motion to avoid these errors.

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