Systems with Nonzero Torque

Edited by Conner Rudzinski Fall 2018

In certain systems, external torques have an effect on a system's angular momentum. Since these external forces do not sum to zero, we end up with a system with non-zero net torque.

The Main Idea

With previous systems involving torque, we've been fortunate enough to have systems where the net torque is non-zero, hence [math]\displaystyle{ \vec{L}_{final} = \vec{L}_{initial}. }[/math]
See systems with zero net torque in "See Also" section below for more information.

However, we're not always fortunate enough to have such systems. In such cases, our computations become a little more complicated, and we'll see how below.

A Mathematical Model

The angular momentum principle is the following: [math]\displaystyle{ {\frac{d\vec{L}}{dt}}= \vec{r} * \vec{F}_{net} = \vec{т}_{net} }[/math]
This means that the calculation of torque is the cross product of the force and distance vectors. We'll now look at the case where [math]\displaystyle{ \vec{F}_{net} }[/math] is NOT equal to 0.

How to Model in VPython

The following code should be self explanatory and can be used as a template for modeling a system involving torque. Important things to note are the use of cross() to calculate the cross product between two vectors and sphere.rotate() to rotate a sphere object around some axis at some angle. One could write their own sub-routine for the cross product and/or rotation, but since vPython comes with these routines, it is advantageous to exploit them.

   # -*- coding: utf-8 -*-
   from __future__ import division
   from visual import *
   
   NUM_LOOP_ITERATIONS = 5000 # Arbitrarily chose 5000
   wheel = sphere(pos = vector(0, 0, 0), radius = 10, color = color.cyan, mass = 5)
   axisOfRotation = vector(5, 0, 0) # Axis of rotation of system
   force = vector(5, 0, 0) # Force acting on system
   delta_t = 1
   t = 0
   angularMomentum= vector(20, 0, 0) # Initial angular momentum
   omega = 40 # Initial angular speed
   inertia = (wheel.mass * wheel.radius ** 2)/12 # Calculating intertia; ML^2 / 12
   dtheta = 0
   while t < 5000:
       rate(500)
       torque = cross(wheel.pos, force) # torque = position x force
       angularMomentum += torque * delta_t # Update angular momentum
       omega = angularMomentum / inertia
       omegaScalar = dot(omega, norm(axisOfRotation))
       dtheta += omegaScalar * delta_t
       wheel.rotate(angle=dtheta, axis = axisOfRotation, origin = wheel.pos)
       t += delta_t

Examples

Simple

If a constant net torque (non-zero) is exerted on an object, which of the following quantities cannot be constant?
A) Moment of inertia
B) Center of mass
C) Angular momentum
D) Angular velocity
E) Angular acceleration

Solution/Explanation:
C, D. Why?
A) Moment of inertia does not change depending on whether torque is exerted on an object; moment of inertia depends on the object and axis of rotation, nothing more.
B) Center of mass doesn't change with applied torque as well.
C) Angular momentum is equal to inertia times angular speed, and we establish below that angular velocity changes.
D) Angular velocity changes since a constant force is being applied to the object, so it's speed/velocity must inherently increase.
E) Because the torque being applied is constant, angular acceleration does not change (remember, acceleration is a measure of the rate of change of velocity!).

System with Non-Zero Torque Calculation

Calculate the torque of a system with a distance vector <2,1,3> and a force vector of <4,3,5>.
This is a system with non-zero torque, so we will use the formula [math]\displaystyle{ \vec{т}_{net} = \vec{r} * \vec{F}_{net} }[/math] The cross product of [math]\displaystyle{ \vec{r} * \vec{F}_{net} }[/math] is [math]\displaystyle{ \vec{2,1,3} * \vec{4,3,5} }[/math], creating the vector [math]\displaystyle{ \vec{-4,2,2} }[/math]. This is the torque vector.

Non-Zero Torque System using magnitude of vectors and Given Angle

Calculate the torque of a system with a distance magnitude of 5 meters and a force magnitude of 7 newtons, working at an angle of 30 degrees.
We will use the formula [math]\displaystyle{ \vec{т}_{net} = \vec{r} * \vec{F}_{net} * sin(theta) }[/math] This calculation becomes 5*7*sin(30) and equals 17.5 Nm.

It is important to recognize that the computation involves a cross product, meaning it is not simply the multiplication of the magnitudes of the two vectors but rather the multiplication of the magnitudes of the two vectors AND the sine of the angle between them. The angle between the two vectors tells us information about how significant the lever arm is in terms of amplifying or decreasing the effect of a force, i.e., leverage.

Middling

Connectedness

How is this topic connected to something that you are interested in?

I had taken physics in high school, and a majority of the concepts covered in this class were familiar to me. However, I had never seen any of the material we covered in the last unit (like angular momentum, conservation of angular momentum, torque, etc.). This material was new and something I found interesting since it went a step further with the information we learned throughout the semester.

Example of physics being used to model the motion of vehicles in Halo 2

How is it connected to your major? Is there an interesting industrial application?

My major is Computer Science, so I cannot really draw a clear line between my current coursework and this topic. That being said, physics is, of course, used in computer science. A good example would be in programming games where most interactions between objects involve physics of some kind, and the programmers/game designers want to model realistic situations in game.

Another example that never really dawned upon me until about halfway through the semester was that both topics require a decent level skill in problem solving/reasoning. I've liked physics this semester because (and I guess this just applies to math in general) it's almost beautiful just to see how, in the end, everything works out and makes sense. Throughout the semester, I've never had a moment during class where I've flat out said, "No, that doesn't make sense" simply because that's never happened. I've never been a huge fan of physics, but even I can appreciate something that comes together so well.

History

Torque is a calculation of the rotational force applied to the system. This concept doesn't have it's own history since it's just a section under torque, so refer to http://www.physicsbook.gatech.edu/Torque#History for more information.

References

Matters and Interactions: 4th Edition
College Physics: Ninth Edition