# How to show how speed varies with incline, and with Gear ratio

• larry1990
In summary: Your model has a constant drag figure, you've ignored air resistance, so you should find there is no top speed for level ground, so I'm not sure where your 2.89 m/s value came from. The net effect of drag and air resistance is that the speed varies with the angle of incline.
larry1990
Hey guys, I'm having difficulty with my university project which is to build an electric bike. My design is to have a friction wheel attached to the shaft of my 800W 2800rpm electric motor which is rated at 2.8Nm. I have two issues with my spreadsheets at the moment. Can someone please help?

Issue 1) I've made a spreadsheet showing how much power is needed to go up various degrees of incline at a constant 5mph. I've decided to design my bike to go 5mph at 1 degrees. I've now been asked to show how speed varies with angle? I'm expecting a stall angle, but my results don't produce it :(. How can I show speed varying with angle whilst also following my initial specification of going up 1 degrees at a constant 5mph? How much speed will this be on a flat 0 degrees? I'm expecting an 'n' shaped graph, but keep getting a 'u' shaped one.

Here is my spreadsheet on various angles and power needed for constant velocity: http://prntscr.com/a77lym
Here is my attempt at speed vs angle: http://prntscr.com/a77lka

Issue 2) Gear ratio. Since the friction wheel will be attached to the shaft of the motor, and it will be driving the rear wheel, the friction wheel will have the same rpm as the motor (2800rpm). My bike wheel is 26inches. If I want the wheel to travel at 5mph. How can I work out my gear ratio? I made a few attempts, but the diameters seem too large. How does torque come into it? Also, if I gear for 5mph, would I be able to go at 10mph with this gearing? or should I change my 5mph specification to 10mph instead?

Here is my attempt at gear ratio: http://prntscr.com/a77mb0

You seem to be completely lost on this .

Show us your workings and in particular show us your diagram of the forces acting on the bike .

For the forces on the bicycle, and verification of power required at different angles to achieve a constant velocity: http://prntscr.com/a7ghpp

For the friction wheel diameter: I assumed the wheel will rotate at the same rpm as the motor shaft (since it's attached t the motor), and assumed it has the same linear velocity as the rear wheel. So v=r(omega); r1(omega)1 = r2(omega)2. Since the rear wheel is 0.33m in radius, and for a linear speed of 2.235m/s, it's 6.77rad/s, and for a rated rpm of 2800 for the motor, it's 293.2rad/s, I find the radius of the friction wheel needed is (6.77 x 0.33)/293.2 = 0.0076m. I'm not sure how to improve on this? I made different selections of gear ratios but I believe that's where I went wrong.

For the analysis of speed vs angle of incline: I used the power required to go up 1 degrees at a constant velocity (as I'm designing to that point), and found the force required for equilibrium (thus constant velocity) down various degrees of incline. Then used this power and divided it by the force to get each corresponding velocity at that angle. But this is wrong because I'm supposed to get a stall angle (a parabola n type curve). But I don't as you have seen.

larry1990 said:
But this is wrong because I'm supposed to get a stall angle (a parabola n type curve). But I don't as you have seen.

Why did you expect a stall angle?
By my rough calculations 152 watts can lift a 90kg mass with 68N of drag directly upward at 0.16 m/s (ie ride up a vertical road..)

larry1990
billy_joule said:
Why did you expect a stall angle?
By my rough calculations 152 watts can lift a 90kg mass with 68N of drag directly upward at 0.16 m/s (ie ride up a vertical road..)

...If you think about it in a practical sense, moving on a flat surface with a constant 152W of power for a constant velocity of 2.89m/s, then climbing an incline without adjusting anything yourself, the speed will reduce to a new constant velocity. Increasing the inclination angle further whilst still using the same power will result in zero velocity at some point (angle). This is why I feel my graph in the screenshot is the exact opposite parabola to what I believe it should be. Or is the imagination here incorrect? Isn't it the same theory with cars going up hills? They need more power when on a hill to avoid stalling?

larry1990 said:
...If you think about it in a practical sense, moving on a flat surface with a constant 152W of power for a constant velocity of 2.89m/s,
Your model has a constant drag figure, you've ignored air resistance, so you should find there is no top speed for level ground, so I'm not sure where your 2.89 m/s value came from. The net force on the bike never goes to zero, the bike will accelerate to infinite velocity..
We know this isn't what will really happens as air drag does exist and is what defines the top speed for pretty much all vehicles (or any objects for that matter) in our atmosphere, including bicycles.

Or is the imagination here incorrect?

Yes.
Power is the rate of doing work, if we have a fixed power source we can still do any quantity of work its just that the rate will vary. For example, your 150 W motor could be mounted in a 10 ton truck, the truck can still climb vertically, it's just at a low rate.

P = Fv

v=P/mg
= 150 W / (10,000 kg * 9.81 m/s2) = 1.5 mm/s
(Though, there are engineering challenges in getting something like this to work)

larry1990
Hey Billy_Joule. Newtons laws tell us we can travel at constant velocity when the net force is zero. The 2.89m/s is a specified constant velocity. I used the power from my table showing the amount of power required to travel up various degrees of incline at a constant speed of 2.235m/s, and reverse engineered it to find the velocity at 0 degrees to provide that 2.235m/s at 1 degrees. I believe it's not the correct value though based on the graph it yielded. Wind resistance, whilst present, in this situation will be minuscule, since you're initially riding at 2.89m/s on a straight. Looking closely at the graph, the velocity reduces a lot when going from 0 degrees to 3 degrees, but then reduces much slower beyond that which is weird.

I used that power and velocity relationship you stated to find the velocities for that graph btw.
Surely, at some angle, the power will not be enough to overcome the drag, and cause a stall (velocity of 0m/s)? The problem with that power equation is that power is a fixed variable here, and therefore the velocity cannot be made to equal 0 through division. This leads me to think the wrong equation is used. Or, another equation is needed in addition to it (i don't know). Do you guys see the issue though?

larry1990 said:
Hey Billy_Joule. Newtons laws tell us we can travel at constant velocity when the net force is zero. The 2.89m/s is a specified constant velocity.
The net horizontal force while on level ground is not zero for any velocity. The bike accelerates to infinity.

larry1990 said:
and reverse engineered it to find the velocity at 0 degrees to provide that 2.235m/s at 1 degrees.
This is not a valid approach at all. Newton laws are well tested, understand those before you try to rewrite the laws of motion. A flat road is fundamentally different to an incline, work is done against gravity in one case only.

I used that power and velocity relationship you stated to find the velocities for that graph btw.
Surely, at some angle, the power will not be enough to overcome the drag, and cause a stall (velocity of 0m/s)?
No.

This leads me to think the wrong equation is used.
No.

Or, another equation is needed in addition to it (i don't know). Do you guys see the issue though?
Yes, the issue was succinctly identified in line one of post #2.
You need to review your notes & textbook/s on Newtons laws and mechanical power.
Don't worry about spreadsheets or graphs yet, they're just muddying the water. build a good understanding of the fundamental concepts and governing equations.

billy_joule said:
The net horizontal force while on level ground is not zero for any velocity.

Are you telling me that at a net force of 0 N, the bike won't travel at a constant speed?

billy_joule said:
Yes, the issue was succinctly identified in line one of post #2.

If I wasn't confused on it, I wouldn't be asking. I Know I've done something wrong, I'm asking for some guidance on how to rectify it.

larry1990 said:
Are you telling me that at a net force of 0 N, the bike won't travel at a constant speed?

No, he's telling you that at a nonzero net power, the net force will not be zero at any speed (unless you account for things like air and rolling resistance).

cjl said:
No, he's telling you that at a nonzero net power, the net force will not be zero at any speed (unless you account for things like air and rolling resistance).

My table does account for rolling resistance. I've used the rolling resistance of rubber on grass (0.06).

## 1. How does incline affect speed?

Incline affects speed by increasing the amount of force required to move an object. As the incline increases, the force of gravity also increases, making it more difficult for an object to move. This results in a decrease in speed.

## 2. Does the gear ratio affect speed?

Yes, the gear ratio has a direct impact on the speed of an object. A higher gear ratio means that the gear connected to the wheel is smaller, resulting in a higher speed. Conversely, a lower gear ratio means a larger gear connected to the wheel and a lower speed.

## 3. How can I show the relationship between incline and speed?

To show the relationship between incline and speed, you can conduct experiments using objects of different masses and inclines of varying degrees. Measure the time it takes for the objects to roll down the incline and record their speeds. You can then plot a graph to visualize the relationship between incline and speed.

## 4. How can I demonstrate the effect of gear ratio on speed?

You can demonstrate the effect of gear ratio on speed by using a bicycle as an example. Have two bicycles with different gear ratios and race them to see which one reaches the finish line first. This will clearly show the impact of gear ratio on speed.

## 5. What factors, other than incline and gear ratio, can affect speed?

Other factors that can affect speed include the surface of the incline, air resistance, and the weight of the object. A smoother surface will result in a faster speed, while air resistance and a heavier object will both decrease speed. Additionally, external forces such as friction can also impact the speed of an object.

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