Gear Ratio in Bicycles using Rotational Motion

[andyrk]

When we change the gears of the bicycle we are riding, we change the the disc we are currently at (which are located at the place where we pedal) to some other disc. This means the radius of the circular disc we were pedaling/rotating changes. So that means if we were rotating the disc with angular velocity $ω$, if $r$ changes (radius of the disc) $rω$ changes. And that means the speed with with the chain which rolls over the disc, i.e. $v = rω$ changes. But how does that make the bike move faster with the same angular velocity we were providing it as before? If we want it to move faster, then the velocity of COM of the rear/front tire should increase, but how does changing gears do that?

 

[A.T.]

...the speed with with the chain which rolls over the disc, i.e. v=rωv = rω changes. But how does that make the bike move faster with the same angular velocity we were providing it as before?

What happens with the angular velocity of the back wheel, if the speed of the chain changes?

 

[andyrk]

Okay..it increases too. But I still didn't get how that increases the speed of the bike? I think it is because of the force imparted by the ground on the rear tire, which happens because the rear tire pushes the ground back..so the ground pushes the tire front..giving it some acceleration. Which increases the velocity of the rear tire and as a result of the entire bike.

Also, if we stop pedaling why does the bike come to a complete halt after sometime? How can it be friction because friction is always acting in the direction of motion of the bike?

 

[Qwertywerty]

Also, if we stop pedaling why does the bike come to a complete halt after sometime? How can it be friction because friction is always acting in the direction of motion of the bike?

Air resistance , friction between parts of the cycle , and other stuff too ( Not sure what that other stuff may be , though ) .

 

[andyrk]

How exactly does the rear tire apply force to the ground? It has a velocity at the bottom-most point..so how does it lead to applying a force?

 

[A.T.]

friction is always acting in the direction of motion of the bike?

No.

How exactly does the rear tire apply force to the ground?

Friction.

 

[andyrk]

No.

Yes.

Friction.

Friction is applied by the ground not by the rear tire. What I am saying is friction is a reaction force. What was the action force?

 

[A.T.]

Friction is applied by the ground not by the rear tire.

Both. There are two equal but opposite forces between tire and ground.

What I am saying is friction is a reaction force. What was the action force?

It doesn't matter which of the two forces you call "action" and which "reaction". They are both friction.

 

[andyrk]

Both. There are two equal but opposite forces between tire and ground.

It doesn't matter which of the two forces you call "action" and which "reaction". They are both friction.

Okay, so I should ask: "How" does the rear tire apply a force on the ground?

 

[Qwertywerty]

Okay, so I should ask: "How" does the rear tire apply a force on the ground?

How or Why ?

 

[andyrk]

Both

 

[A.T.]

"How" does the rear tire apply a force on the ground?

The horizontal component is friction.
https://en.wikipedia.org/wiki/Friction#Static_friction
https://en.wikipedia.org/wiki/Friction#Kinetic_friction

 

[Qwertywerty]

When angular velocity of the back wheel increases , the wheel gains a tendency to slip on the ground . Friction opposes this tendency of relative motion by increasing - this causes acceleration of the wheel's center of mass , causing increase in velocity of the Com , thus making net velocity of point in contact with ground zero , thus preventing relative motion , and in this process causing increase in bike's speed .

 

[A.T.]

When angular velocity of the back wheel increases , the wheel gains a tendency to slip on the ground . Friction opposes this tendency of relative motion by increasing - this causes acceleration of the wheel's center of mass , causing increase in velocity of the Com , thus making net velocity of point in contact with ground zero , thus preventing relative motion , and in this process causing increase in bike's speed .

If there is no slippage, then angular velocity and COM's linear velocity change simultaneously. So I don't see how you can tell which change is causing the other change. Seem arbitrary to me.

 

[Qwertywerty]

If there is no slippage, then angular velocity and COM's linear velocity change simultaneously. So I don't see how you can say which change is causing the other change. Seem arbitrary to me.

Well the angular velocity increases with time , and so it must be causing simultaneously , the velocity of Com to increase too .

Take a rubber ball , spin it and drop on the ground . It would first slip and then eventually , start to roll . What would you say is the mechanism/cause for this ?

 

[A.T.]

Well the angular velocity increases with time , and so it must be causing simultaneously , the velocity of Com to increase too .

You could just as well say:

The velocity of Com increases with time, and so it must be causing simultaneously , the angular velocity to increase too .

Take a rubber ball , spin it and drop on the ground . It would first slip and then eventually , start to roll

But here the angular velocity would decrease, the opposite of the case above. And a bike isn't dropped with spinning wheels, so what does it have to do with an accelerating bike?

 

[Qwertywerty]

You could just as well say:

The velocity of Com increases with time, and so it must be causing simultaneously , the angular velocity to increase too .

So changing the gear doesn't cause a direct increase in angular velocity of the wheel ?

But here the angular velocity would decrease, the opposite of the case above. And a bike isn't dropped with spinning wheels, so what does it have to do with an accelerating bike?

My point here is that friction , in the ball's case is opposing the relative motion and therefore causing an increase in ball's horizontal velocity ; and the only difference in the case of the ball and wheel is that the ball had slipping in the start and hence friction opposed angular velocity , in the case of the wheel it would not , it would merely increase the Com's velocity .

 

[A.T.]

So changing the gear doesn't cause a direct increase in angular velocity of the wheel ?

Switching gears doesn't necessary change the bike speed, which (assuming no slippage) determines the angular velocity of the wheel.

My point here is that friction , in the ball's case is opposing the relative motion...

You can (and usually do) accelerate a bike without any relative motion at the contact (slippage).

 

[Qwertywerty]

Switching gears doesn't necessary change the bike speed, which (assuming no slippage) determines the angular velocity of the wheel.

No - I don't think so either . But the case the OP talks about involves increasing the speed of the bike .

You can (and usually do) accelerate a bike without any relative motion at the contact (slippage).

That is what I have been saying all along . What I've been trying to do is give the reason for this .

 

[A.T.]

You can (and usually do) accelerate a bike without any relative motion at the contact (slippage).

What I've been trying to do is give the reason for this .

Locally the "reason" is static friction. Globally the "reason" is simply that the equations of motion say so, given certain boundary conditions.

 

[Qwertywerty]

Locally the "reason" is static friction.

Would you care to explain how this works ?

 

[A.T.]

Would you care to explain how this works ?

https://en.wikipedia.org/wiki/Friction#Static_friction

Also several thread on this forum discuss this.

 

[Qwertywerty]

Also several thread on this forum discuss this.

I know how friction works , I'm asking what the mechanism is in this case - Angular velocity increases , so ...

 

[A.T.]

I'm asking what the mechanism is in this case - Angular velocity increases , so ...

It's your claim, not mine, that the increase of angular velocity is somehow a fundamental cause that triggers the rest.

 

[Qwertywerty]

It's your claim, not mine, that the increase of angular velocity is somehow a fundamental cause that triggers the rest.

Ok then . Do have some explanation to share ?

 

[A.T.]

Ok then . Do have some explanation to share ?

Not beyond what I already wrote in post #20.

 

[SteamKing]

Also, if we stop pedaling why does the bike come to a complete halt after sometime? How can it be friction because friction is always acting in the direction of motion of the bike?

You've got things backwards here. Friction acts to retard motion. It may not be the friction you are thinking of in the contact between the tire and the ground, but what about in the wheel and axle bearings?

 

[russ_watters]

I don't get why this question keeps coming up. People seem to have a tendency to way overthinking what a bike does.

Changing gears (on anything) does not automatically increase the speed of the device. The thing that changes almost instantly is the rpm of the motor (for a bike, your legs).

Now, if you gear-up, slowing down your legs, but reducing your mechanical advantage (less torque transmitted to the wheels, fractionally), how does this help you accelerate...?

 

[andyrk]

How does a faster spinning rear wheel generate more push from the ground?

 

[A.T.]

How does a faster spinning rear wheel generate more push from the ground?

Who says it does?

 

[andyrk]

If it doesn't then why does changing gears make the rear wheel go faster (COM)?

 

[A.T.]

why does changing gears make the rear wheel go faster (COM)?

It doesn't necessarily.

 

[andyrk]

Then what does changing gears even do?

 

[A.T.]

Then what does changing gears even do?

It changes the ratios of angular velocities and torques, between back wheel and pedals.

 

[andyrk]

Which would do what?

 

[russ_watters]

Then what does changing gears even do?

Did you not read my post #28?

(Or, to answer a different way: exactly what AT just said and nothing more.)

 

[andyrk]

I don't understand why the rear wheel accelerates more, by reducing the rpm of our legs/where we pedal.

 

[SteamKing]

I don't understand why the rear wheel accelerates more, by reducing the rpm of our legs/where we pedal.

You seem to be hazy on the concept of mechanical advantage.

Bikes use different gears to reduce the rider's effort at maintaining speed.

By using his feet on the pedals, the bike rider creates a torque which is transmitted by the chain to the rear wheel. This torque is what turns the wheel, which then moves the bike, and makes it go. If you want to go fast, it stands to reason that the rear wheel must turn fast, which in turn means the rider must pedal more quickly. This gets rather exhausting, even for competition riders. What if there were some way the rider could keep turning the pedals with his feet at the same rate all the time, but use a special device to make the rear wheel turn faster?

Fortunately, there is: the gear.

Study this article on gears:

https://en.wikipedia.org/wiki/Gear

 

[sophiecentaur]

Velocity in the frame of the ground is actually zero - unless you have wheel spin.

It has a velocity at the bottom-most point

 

[russ_watters]

I don't understand why the rear wheel accelerates more, by reducing the rpm of our legs/where we pedal.

Have you pedaled a bike before? What happens to your ability to apply force to the pedals as the rpm gets too high? Or: is it easier to apply a large force when your legs are moving fast or slow?

 

[A.T.]

I don't understand why the rear wheel accelerates more, by reducing the rpm of our legs/where we pedal.

I don't understand why you think this must be the case.

 

[russ_watters]

We really try to avoid spoonfeeding here because people learn better (and learn how to learn better) if they figure things out on their own after being given a gentle push in the right direction. But you're just not getting it, so I'm going to lay it out for you:

A bicycle begins to accelerate (from a stop or a constant speed) because more torque is applied to the back wheel. To get more torque to the back wheel, you push harder on the pedals.

The reason bikes are geared is because there is a limited range of speeds at which a human can push on a pedal. Above perhaps 120 rpm, our legs simply can't move fast enough to keep pushing on the pedals - you spend most of your effort basically "chasing" the pedal with your foot. So gearing-up allows you to push harder on the pedals, by lowering their rpms.

 

[sophiecentaur]

I don't understand why the rear wheel accelerates more, by reducing the rpm of our legs/where we pedal.

I get the feeling that you have stopped thinking about what your actual experience has been of riding a bike and latched yourself in a totally 'verbal' mode of approach. You expect too much if you insist on an answer that fits totally into your requirements for an 'explanation'.
The general topic of levers and gears is discussed all over the place (internet and books) and there are countless examples of Force Magnifiers and Velocity Magnifiers, where the terms Velocity Ratio and Mechanical Advantage are explained and discussed. You have to make a personal effort with this and not rely on other people to find the magic bullet to solve your problem.

It could be that you have read something, somewhere and you are trying to fit what you read or understood from it to the answers you have been getting on PF. It may be that your memory or interpretation has failed you and you should start again with another source (e.g. this thread).

 

[andyrk]

A bicycle begins to accelerate (from a stop or a constant speed) because more torque is applied to the back wheel

So what does more torque make the rear wheel do? Make it spin faster?

 

[Chestermiller]

So what does more torque make the rear wheel do? Make it spin faster?

The increased torque at the pedals is delivered to the rear wheel by the chain, which exerts an increased torque on the rear gear. The rear gear is rigidly affixed to the wheel. So the increased torque on the rear gear is accompanied by an increased torque on the rear wheel. This causes it to exert a larger backward force on the ground, which, in turn, exerts the larger forward force on the wheel. This larger forward force is transmitted through the wheel to the axle, and from that, to the frame. This causes the body of the car to accelerate. The increasing linear velocity of the body is accompanied by an increasing rate of rotation of the wheel.

It is possible to model all this pretty easily. Such a model would include the rear wheel, the rear gear, the chain, the bar joining the rear wheel axle to the pedal axle, the pedal gear, and the pedal. Such a model would enable one to quantify the effect of different gear diameters on (a) the ratio of the pedal angular velocity to the rear wheel angular velocity and (b) the ratio of the imposed pedal torque to the forward force exerted by the ground on the rear tire.

Chet

 

[andyrk]

Amazing. I understood everything now.

 

[russ_watters]

So what does more torque make the rear wheel do? Make it spin faster?

Yes. And because of friction with the ground, instead of just spinning free and slipping, it accelerates the bike forward as it spins faster.

 

[andyrk]

Also, when we change gears, why does it become easier to pedal(less effort, pedals feel lighter) or difficult to pedal (more effort, pedals feel heavier), depending on the gear we shifted to? Does this have any physics related to it?

 

[A.T.]

Also, when we change gears, why does it become easier to pedal(less effort, pedals feel lighter) or difficult to pedal (more effort, pedals feel heavier), depending on the gear we shifted to? Does this have any physics related to it?

Have you looked into the physics of simple levers? Gears are levers working continuously.

 

[andyrk]

Have you looked into the physics of simple levers? Gears are levers working continuously.

What does it have to do with physics?