# Why does torque increase as we increase distance from the centre axis?

• Frigus

#### Frigus

Earlier I read that as distance from centre of axis of rod increases it's inertia increases but why does torque increases isn't it like that if inertia increases for same force the less force should be produced?

Earlier I read that as distance from centre of axis of rod increases it's inertia increases but why does torque increases isn't it like that if inertia increases for same force the less force should be produced?

Force is not produced; force is applied.

Can you describe more clearly the scenarion you have in mind?

Force is not produced; force is applied.

Can you describe more clearly the scenarion you have in mind?
I want to say for same force why does more rotational effect produce

I want to say for same force why does more rotational effect produce
Are you familiar with mechanical advantage and as it applies to levers?

Are you familiar with mechanical advantage and as it applies to levers?
Yes sir

Yes sir
That's torque!

• Bystander, anorlunda and jbriggs444
I want to say for same force why does more rotational effect produce

One way to understand this is through energy. Imagine a uniform rod of length ##2L## fixed at the centre:

1) Apply a force ##F## at the end of the rod through one revolution.

2) Apply a force ##F## a distance ##L/2## from the centre through one revolution.

Then, look at the work done by the force in each case:

1) ##W_1 = Fd_1 = F(2\pi L) = 2\pi FL##

2) ##W_2 = Fd_2 = F(2 \pi L/2) = \pi FL##

The work done by the force in the first case is twice that done by the force in the second case. The rod, therefore, has more rotational kinetic energy (RKE) in the first case.

We can then use the energy equation to find the equation for angular acceleration and angular momentum. I can leave that as an exercise for you.

• russ_watters
why does torque increases isn't it like that if inertia increases for same force the less force should be produced?

With torque, it's the place where the force is applied. But with rotational inertia it's the place where mass is located. You can change either of those places without having any effect on the other placement, so they are independent of each other. That is is, you can change the torque with it having no effect on the rotational inertia, and vice-versa.

Maybe we can ask the question in a different way:

Why did physicists invent or define a concept that depends on the applied force, but also depends on where that force is applied?

The answer is, it comes from our daily physical experience when we try to rotate something that does not want to rotate. Let's say we want to loosen a bolt with a spanner. We do find that if we apply a force farther away from the bolt, then the same force is better at loosening the bolt. [* see note below]

When we do some careful experiments, we find that (force applied to spanner) X (distance from bolt) is a useful number that can tell us how hard it is to turn that bolt, because it takes the mechanical advantage into account. Once we calculate that number, we can use it to describe the problem in a more general way -- and someone else with a different spanner can find out how much force they would have to apply.

So they just invented this useful product of two numbers and called it torque because it helps to describe a lot of situations where someone is trying to rotate something.

[*] But why is the far-away force better than a force near the bolt? You already know about mechanical advantage, so I can just say, "because of mechanical advantage". But how does mechanical advantage actually work? I will try to post a better explanation here soon.

Sometimes I find that it's better to think about something intuitively without worrying about how to calculate it. After we have the basic picture clear, we can think about formulas.

So think about a bridge that is supported on two pillars -- one pillar at each end of the bridge. A bus starts from one end of the bridge and moves towards the other end. When the bus is right over the first pillar, that pillar has to support all of the weight, and the other pillar doesn't have to support any weight. As the bus moves over the bridge, the first pillar has to support less and less weight and the second pillar has to support more and more of the weight of the bus. When the bus is at the middle of the bridge, each pillar has to support half the weight of the bus.

Now think of the bridge as a lever. Think of the first pillar as the fulcrum. Imagine that the second pillar is actually someone ( a giant ) who is trying to support the bus using that lever. And think of when the bus is ##1 / 10## of the way from the first pillar. Clearly, the giant gets a big mechanical advantage from this lever, because the fulcrum (the first pillar) is taking most of the weight.

(Of course, we are pretending here that the bridge itself has no weight compared to the bus).

As I said, we can worry about the formula later, after we get the intuitive picture from the above.

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• Delta2
You might ask yourself why the pikes in this picture are long. 'They' didn't invent torque before there was a need to. No one said "I know, let's take Force times Perpendicular Distance and call it Torque"
What was found was that long levers need less force when applied against short levers. (Obvs, of course) An empirical rule was postulated and then confirmed by countless measurements and observations. The principle of Moments was born. Torque was the word which described the effect of levers.

I don't think there can be any actual proof of this - any attempt would involving just re-stating an idea and then re-stating it in a different way - proving nothing. It's just like Momentum (another quantity that we have found to be always conserved (when you get rid of all the practicalities of an experiment)

In the case of many so-called Machines, the geometry of the thing gives a theoretical 'Velocity Ratio' and that's often confused (even PF is incredibly sloppy about this) with 'Mechanical Advantage', which is the measured ratio of load one effort. It always falls short of VR because of Inefficiency. People ignore the dead weight of lever arms, friction and the weight of pulley blocks. The MA of that Capstan in the above picture will be hopelessly worse than the VR might suggest.

• Frigus
The MA of that Capstan in the above picture will be hopelessly worse than the VR might suggest.
The MA of that particular capstan is zero because there is no load. No rope is attached. The tourists are busy doing nothing but turning the dead weight.

• russ_watters, Swamp Thing and sophiecentaur
'They' didn't invent torque before there was a need to. No one said "I know, let's take Force times Perpendicular Distance and call it Torque"
What was found was that long levers need less force when applied against short levers. (Obvs, of course) An empirical rule was postulated and then confirmed by countless measurements and observations. The principle of Moments was born. Torque was the word which described the effect of levers.

I don't think there can be any actual proof of this - any attempt would involving just re-stating an idea and then re-stating it in a different way - proving nothing. It's just like Momentum (another quantity that we have found to be always conserved (when you get rid of all the practicalities of an experiment)

In the case of many so-called Machines, the geometry of the thing gives a theoretical 'Velocity Ratio' and that's often confused (even PF is incredibly sloppy about this) with 'Mechanical Advantage', which is the measured ratio of load one effort. It always falls short of VR because of Inefficiency. People ignore the dead weight of lever arms, friction and the weight of pulley blocks. The MA of that Capstan in the above picture will be hopelessly worse than the VR might suggest.

Thanks sir,
Your explanations always found to be very intuitive to me.

And can you help me to get out of this problem

when we say something is directly proportional to something then why do we multiply the to directly proportional terms for example if I say x directly proportional to y and I also say that x is also directly proportional to z then while writing the whole why do we say that the x is directly proportional to y z

The MA of that particular capstan is zero because there is no load. No rope is attached. The tourists are busy doing nothing but turning the dead weight.
Their eyes would certainly be bulging a bit more if a chain and anchor were attached. (They are working with an MA of 0!)

when we say something is directly proportional to something then why do we multiply the to directly proportional terms for example if I say x directly proportional to y and I also say that x is also directly proportional to z then while writing the whole why do we say that the x is directly proportional to y z
Instead of asking why we multiply, let us examine a simpler question: Does multiplication work?

Take, for instance, the formula for gravitational force:$$F=G\frac{m_1 m_2}{r^2}$$
Is that formula for F directly proportional to ##m_1##?
Is that formula for F directly proportional to ##m_2##?
Is that formula for F inversely proportional to ##r^2##?

If so then what is the problem? The formula works, so use it.

Instead of asking why we multiply, let us examine a simpler question: Does multiplication work?

Take, for instance, the formula for gravitational force:$$F=G\frac{m_1 m_2}{r^2}$$
Is that formula for F directly proportional to ##m_1##?
Is that formula for F directly proportional to ##m_2##?
Is that formula for F inversely proportional to ##r^2##?

If so then what is the problem? The formula works, so use it.
Sir but i want to know why we multiply these terms not added nor subtracting why we multiply these without any reason.

Sir but i want to know why we multiply these terms not added nor subtracting why we multiply these without any reason.
But there is a reason. It's defined that way. And the reason it's defined that way is because it's useful. When something is in rotational equilibrium (that is, it's not rotating or it's rotating at a steady speed) it's always found that the net torque is zero. If you calculate torque as force times lever arm you find that the net torque is zero. If you define torque some other way then you may find that the net torque is not zero, and so torque is not useful for determining rotational equilibrium.

This is the way it goes in science and technology. People invent quantities and if those quantities are useful then others use them and they make their way into the textbooks and students are asked to learn them. Quantities that are not useful get discarded and do not appear in textbooks.

But there is a reason. It's defined that way. And the reason it's defined that way is because it's useful. When something is in rotational equilibrium (that is, it's not rotating or it's rotating at a steady speed) it's always found that the net torque is zero. If you calculate torque as force times lever arm you find that the net torque is zero. If you define torque some other way then you may find that the net torque is not zero, and so torque is not useful for determining rotational equilibrium.

This is the way it goes in science and technology. People invent quantities and if those quantities are useful then others use them and they make their way into the textbooks and students are asked to learn them. Quantities that are not useful get discarded and do not appear in textbooks.

So sir it is like that scientists done hit and trial and make formulas by guessing And try to do application of it and if they get successful then we use that equation.

when we say something is directly proportional to something then why do we multiply the to directly proportional terms for example if I say x directly proportional to y and I also say that x is also directly proportional to z then while writing the whole why do we say that the x is directly proportional to y z
The axioms and terms of arithmetic are well established. I am not totally clear about your notation here but I think your final statement could be wrong or incomplete. Simple proportionality is between two variables. If z is constant then it is a constant of proportionality and there is still a 'straight line graph' for x against y. If that is not made clear and (say) z is a function of x, the proportionality no longer holds.
If you want to argue about the basics of Maths then get a good textbook first and make sure you are sure of the subject step by step and that you are clear about the arithmetical rules. They work very well (perfectly) if you also use the right notation without the 'words'.

So sir it is like that scientists done hit and trial and make formulas by guessing And try to do application of it and if they get successful then we use that equation.
Yep. That's very often the way 'laws' of nature are found to apply. Everything has to be verified by measurement in the end after intelligent guesses about the probable maths involved. This is one of the problems that people have with String Theory; we just don't have the necessary equipment to test them so String Theory should perhaps be called String Hypothesis or String Conjecture.

• weirdoguy
So sir it is like that scientists done hit and trial and make formulas by guessing And try to do application of it and if they get successful then we use that equation.
Very often, yes.

• Swamp Thing
The MA of that particular capstan is zero because there is no load. No rope is attached. The tourists are busy doing nothing but turning the dead weight.
"Yo, ho, ho, me hearties! And a big, big welcome to this nautical version of the dude ranch! Heave away with a will, now, me lubbers, if you don't want to feel the end o' this cat o' nine tails!"

• sophiecentaur
Sir but i want to know why we multiply these terms not added nor subtracting why we multiply these without any reason.

Sir but i want to know why we multiply these terms not added nor subtracting why we multiply these without any reason.
Suppose you spent $2 for coffee every day for 30 days. How much money did you spend on coffee in total? Is it 2*30=$60 or 2+30=\$32? What is the reason?

• sophiecentaur and jbriggs444
One way to understand this is through energy. Imagine a uniform rod of length ##2L## fixed at the centre:

1) Apply a force ##F## at the end of the rod through one revolution.

2) Apply a force ##F## a distance ##L/2## from the centre through one revolution.

Then, look at the work done by the force in each case:

1) ##W_1 = Fd_1 = F(2\pi L) = 2\pi FL##

2) ##W_2 = Fd_2 = F(2 \pi L/2) = \pi FL##

The work done by the force in the first case is twice that done by the force in the second case. The rod, therefore, has more rotational kinetic energy (RKE) in the first case.

We can then use the energy equation to find the equation for angular acceleration and angular momentum. I can leave that as an exercise for you.
Thanks sir I understood it. 