What is the relationship between torque and F=ma in rotational motion?

[Storm Butler]

Hello,
The concept of torque has always been confusing to me for a few reasons, but i guess it boils down to two things that really seem to bother me:
1.) We all posses an intuitive sense of the fact that pushing on a lever further away from the fulcrum makes the pushing easier. Most of the time the concept of torque is started off from this "intuition". I was wondering if anyone has worked out the equation for torque just starting off with F=ma and not using any intuition. Personally when i think about it using F=ma i can't think of why pushing on a door should be easier further away from the hinges, or that the proptionality should be a linear factor. Also i don't see how one comes up with the expression r$_{x}$F$_{y}$-F$_{x}$r$_{y}$ should show up using F=ma.

2.) When analyzing a gyroscope textbooks seem to throw around these concepts of torque, angular momentum, and moment of inertia. With all the right hand rules and tensors. I guess, once again i want to know if anyone knows how to work out this analysis not using torques and the such; just using F=ma. For example when the gyroscope starts to tilt on the axis it doesn't fall down, it just perceses. Where is the force pushing against gravity come from?

Thanks for any help.

 

[cepheid]

Hello,
The concept of torque has always been confusing to me for a few reasons, but i guess it boils down to two things that really seem to bother me:
1.) We all posses an intuitive sense of the fact that pushing on a lever further away from the fulcrum makes the pushing easier. Most of the time the concept of torque is started off from this "intuition". I was wondering if anyone has worked out the equation for torque just starting off with F=ma and not using any intuition. Personally when i think about it using F=ma i can't think of why pushing on a door should be easier further away from the hinges, or that the proptionality should be a linear factor. Also i don't see how one comes up with the expression r$_{x}$F$_{y}$-F$_{x}$r$_{y}$ should show up using F=ma.

I think you have a little bit of confusion about how physical quantities arise in physics. Torque is the way it is, because we define it to be that way. Why do we use quantities with the particular definitions that they have, and not some other definitions? Because the definitions that we have chosen to use are useful in some way, and allow us to mathematically model physical situations. Torque is the rate of change of angular momentum with time. Angular momentum is a really useful quantity for describing the rotational motion of an object, because it is a conserved quantity.

The whole point is that F = ma is not sufficient for describing rotational motion. For instance, how hard it is to change the state of rotational motion of a body depends not only on the mass of that body, but also on how that mass is distributed around the axis around which you're trying to rotate it. That's why the concepts of moment of inertia, torque, etc. are required to deal with this situation.

2.) When analyzing a gyroscope textbooks seem to throw around these concepts of torque, angular momentum, and moment of inertia. With all the right hand rules and tensors. I guess, once again i want to know if anyone knows how to work out this analysis not using torques and the such; just using F=ma. For example when the gyroscope starts to tilt on the axis it doesn't fall down, it just perceses. Where is the force pushing against gravity come from?

Thanks for any help.

There is no force pushing against gravity. It's just that gravity has a different effect on the spinning object than it does on an object that does not have any angular momentum. If you just try to understand this using Newton's laws for translational motion, you will fail. To understand why this is the case, you need to make use of concepts from rotational dynamics like the ones you listed above.

 

[CWatters]

Read up on levers? When you push a door open mass near the hinge is operated on by a lever.

The moment of inertia of something like a cylinder can be calculated by considering it as lots of little masses at different distances from the pivot and then integrating.

 

[Storm Butler]

I think i may have explained my hold up incorrectly. Its not that I don't believe the equations work or that i don't realize that the mathematics of physics is just to model physical behavior. Rather I am confused how some one even thought up these expressions. Obviously there are extra terms added to our Newtonian equations in non-inertial refrence frames. However, that doesn't mean it can't be explained through our linear Newtonian concepts. For example the centripetal acceleration may just look like a strange term and maybe you just accredited to the fact that the frame is non-inertial but we can look at an actuall example like an orbiting object. You may ask why the object doenst fall like an a ball near the Earth but you can show that it does. you could see how fast it is moving and show that the amount of distance the object travels and the amount it falls down is exactly compensated for through the curvature of the earth.

same thing for torque. I feel like if i were to sit down with a sufficiently intellgent person and say i know things rotate and i know F=ma and all the other kinematic equations then i should be able to come up with some equations that describe this rotating motion and it should match up with torque. Unfortanetly i don't see how to do this.

same thing for the levers. obviously the equations work and they even make sense based on our intuition. by how do they get derived? you can't just say levers are easier to move further from the fulcrum and it obeys these equations without getting to those equations using Newtons axioms.

 

[AdkinsJr]

The fact that levers are easier to move further from the fulcrum was known before Newton.

http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesLever.shtml

You don't have to derive from general principles to know or mathematically describe physics. I asked the same question in physics when we studied magnetic fields, the force on a charged particle with a velocity v in the magnetic field B was the cross product $$F=qV× B$$ which can't really be derived from more general principles either, it is what we observe.

 

[DrewD]

I think you can derive torque from conservation of energy by looking at the work done. The farther from the pivot point, the farther the force moves the object and therefore must be a lesser force than would need to be applied nearer to the pivot. Otherwise, we could get free energy.

I find that illuminating, but I don't think it is really any more fundamental than just observing the way things move.

 

[Storm Butler]

The fact that levers are easier to move further from the fulcrum was known before Newton.

I know that the concept of levers "reducing" the amount of force needed to move things was known before Newton. I would even feel safe in saying people understood this before archimedes. Rather i was wondering how to show this in the context of Newton's Physics (i.e. using his axioms).

For example, you mentioned the lorentz force on a charge which has a cross product in it. It's true that, that this equation agrees with our observations of a charge in a field. (maybe it was even formulated from these observations I am not really sure) but you can also come up with it using maxwells equations and the lagrangian formulation of physics. So in this case we have a purely theoretical derivation albiet a bit abstract one.

Or you can consider the corriolis effect which also involves the cross product but whose physical origin can understood intuitively through the geometry and motion of the system.

Drew D, the thought about using conservation of energy is interesting. Perhaps this is what i was looking for, Ill have to fool around with it a bit. Still i would find it more satisfying if there could be an analysis done using F=ma and looking at the system of the rotating objects.

 

[Muphrid]

We define a quantity called angular momentum, and for a pointlike object with linear momentum $p$, the angular momentum of that object about a given fixed point $r_0$ is

$$\ell = (r - r_0) \times p$$

So far, all we've done is define a quantity. Now, we can look at time derivatives of that quantity. We say that the angular momentum, position, and linear momentum may all vary with time, so that

$$\frac{d\ell}{dt} = \frac{dr}{dt} \times p + (r - r_0) \times \frac{dp}{dt}$$

But clearly $dr/dt = v$ and $v \times p = 0$. On the other hand, $dp/dt = F$, the force, so we have

$$\frac{d\ell}{dt} = (r - r_0) \times F$$

We often take $r_0 = 0$ for simplicity. Hence, with $F = ma = dp/dt$, we're able to derive the law of conservation of angular momentum. We define the word torque to mean "the time derivative of angular momentum," just as we define force to mean "the time derivative of linear momentum".

 

[BruceW]

The whole point is that F = ma is not sufficient for describing rotational motion. For instance, how hard it is to change the state of rotational motion of a body depends not only on the mass of that body, but also on how that mass is distributed around the axis around which you're trying to rotate it. That's why the concepts of moment of inertia, torque, etc. are required to deal with this situation.

same thing for torque. I feel like if i were to sit down with a sufficiently intellgent person and say i know things rotate and i know F=ma and all the other kinematic equations then i should be able to come up with some equations that describe this rotating motion and it should match up with torque. Unfortanetly i don't see how to do this.

It does all match up. You can explain all rotational motion using only F=ma. (And for continuous medium, you simply need to take the limit of taking lots of small masses). So you can derive all the properties of the concepts of torque, angular momentum, e.t.c. from using F=ma. In other words, you could get by without ever talking about torque and angular momentum. BUT obviously it saves a lot of time in many engineering situations to be able to use the concepts of torque, e.t.c.

Um, also you probably need to use Newton's 3rd law between the masses to explain the concepts of torque, e.t.c. But you can see what I mean, that these concepts can be explained.