# Biking up a hill in different gears

## Homework Statement

Hi there,
I am doing a project for my physics class where I am attempting to calculate the amount of force and work required to go up a 10% grade hill in a variety of different gears and I am running into a lot of issues. I have already collected my data and have calculated gear ratios for each of my gears as well as velocity for my different gears going up the hill. What I am wondering is, how do you calculated the force and work needed to go up a hill in a certain gear? I understand that gravity and normal force are at play as well as frictional force but I am having trouble getting the force applied by the rider to the pedal and getting my head around everything. If you need more information I have it so please ask.
Thanks!

## The Attempt at a Solution

billy_joule
how do you calculated the force and work needed to go up a hill in a certain gear? I understand that gravity and normal force are at play as well as frictional force

You need a good free body diagram. Start by finding an expression for Fwheel (the force the driven wheel applies to the road) in terms of θ (angle of incline), v (bike velocity - parallel to the incline), g, m (mass of rider & bike), Frolling resistance and Fair drag. If that's too difficult, try again without Frolling resistance and Fair drag.

Start by finding an expression for Fwheel (the force the driven wheel applies to the road) in terms of θ (angle of incline)

How do you calculate the Force of the driven wheel?

billy_joule
How do you calculate the Force of the driven wheel?
From the free body diagram, are you familiar with that term? (try googling 'bicycle free body diagram')
If the bike is at constant velocity then acceleration is zero, so, from newtons second law, all the forces on it must sum to zero:
ΣF = ma = 0

we know there is a component of gravity force acting down the incline (this is why we can bike down a hill without pedaling), so, a free body diagram will show that to bike up the hill the bike must exert a force which is equal in magnitude and opposite in direction to the force of gravity parallel to the incline (ignoring friction for now).

Yes! I know free body diagrams well and I understand the concepts you are mentioning. The only part I am confused on is the actual calculations and formula of finding the force needed by the biker to summit the hill (I assume thats what the drive force is?)

From the free body diagram, are you familiar with that term? (try googling 'bicycle free body diagram')
If the bike is at constant velocity then acceleration is zero, so, from newtons second law, all the forces on it must sum to zero:

ΣF = ma = 0

we know there is a component of gravity force
acting down the incline (this is why we can bike down a hill without pedaling), so, a free body diagram will show that to bike up the hill the bike must exert a force which is equal in magnitude and opposite in direction to the force of gravity parallel to the incline (ignoring friction for now).
Sorry forgot to reply to you directly.

billy_joule
Yes! I know free body diagrams well and I understand the concepts you are mentioning.

So once you have a free body digram of the bike as a whole, and so know the force the drive wheel applies to the road then adding detail to that free body diagram to include the internal forces of the bike will find the force applied to the pedal. The pedal force will be in terms of the wheel force, the wheel radius, the gear ratio and the crank radius.

So once you have a free body digram of the bike as a whole, and so know the force the drive wheel applies to the road then adding detail to that free body diagram to include the internal forces of the bike will find the force applied to the pedal. The pedal force will be in terms of the wheel force, the
I think im slowly getting the hang of it! :) how do we find the the force wheel? I understand how you're getting to the force applied to the pedal which is ultimately the thing I am going to solve for with all my data so I can find out what gear is best for which hill!

billy_joule
I think im slowly getting the hang of it! :) how do we find the the force wheel?
Posts #2 and #4 attempted to explain that, show your working and we can see where exactly you are stuck.

I understand how you're getting to the force applied to the pedal which is ultimately the thing I am going to solve for with all my data so I can find out what gear is best for which hill!
Unfortunately physics alone can't answer that question! You can find an upper and lower limit using some common sense but you need to turn to biomechanics to get any sort of accurate answer. Fortunately there's quite a bit of info around in the topic, try googling "cycling cadence efficiency" or "optimal cycling cadence". The human body as an engine has, like most engines, has a particular operating point where it generates the most power, to get up the hill the fastest you need to be pedalling at whatever cadence your peak power output is.

Posts #2 and #4 attempted to explain that, show your working and we can see where exactly you are stuck.

Unfortunately physics alone can't answer that question! You can find an upper and lower limit using some common sense but you need to turn to biomechanics to get any sort of accurate answer.
Fortunately there's quite a bit of info around in the topic, try googling "cycling cadence efficiency" or
"optimal cycling cadence". The human body as an engine has, like most engines, has a particular

operating point where it generates the most power, to get up the hill the fastest you need to be pedalling at whatever cadence your peak power output is.

The only part I don't understand is how to find the force wheel. How does it relate to the angle?

Secondly, on post 7 you mention the pedal force would be related to the radius of the rear wheel and crank etc. Does that mean they are multiplied? The problem I'm having now isnt with the concepts you're mentioning it's with the actual formulas and equations

Nidum
Gold Member
• billy_joule
billy_joule
The only part I don't understand is how to find the force wheel. How does it relate to the angle?

Secondly, on post 7 you mention the pedal force would be related to the radius of the rear wheel and crank etc. Does that mean they are multiplied? The problem I'm having now isnt with the concepts you're mentioning it's with the actual formulas and equations
Well, the equations should appear from your free body diagrams, if the concepts are understood then the equations should follow.Your questions imply you have an incorrect or incomplete FBD. Like I said, show your work. Generally, a correct free body diagram is the hard part, the rest should be plug and chug.
Check out Nidiums link, it should help a lot.

Well, the equations should appear from your free body diagrams, if the concepts are understood then the equations should follow.Your questions imply you have an incorrect or incomplete FBD. Like I said, show your work. Generally, a correct free body diagram is the hard part, the rest should
be plug and chug.
Check out Nidiums link, it should help a lot.
I really just need the equations right now .

CWatters
Homework Helper
Gold Member
I am doing a project for my physics class where I am attempting to calculate the amount of force and work required to go up a 10% grade hill in a variety of different gears and I am running into a lot of issues.

Would the choice of gear effect the work required to climb a hill? What is the gain in Potential Energy?

CWatters
Homework Helper
Gold Member
I really just need the equations right now .

Take billy's advice and sketch the FBD. You might be surprised how fast you can work out the equation from the FBD.

Would the choice of gear effect the work required to climb a hill? What is the gain in Potential Energy?
It would I believe. The gain in potential energy would in theory be the same tho

CWatters
Homework Helper
Gold Member
The work required to climb the hill is just the change in PE.

Any other work expended must be due to either a change in KE (faster at the top vs the bottom) or losses due to friction or air resistance. If you went up at a reasonably constant speed then the change in KE can be ignored. If you went up reasonably slowly in all gears you might also be able to ignore air resistance.

The work required to climb the hill is just the change in PE.

Any other work expended must be due to either a change in KE (faster at the top vs the bottom) or
losses due to friction or air resistance. If you went up at a reasonably constant speed then the change in KE can be ignored. If you went up reasonably slowly in all gears you might also be able to ignore
air resistance.

How do I calculate the PE?

CWatters