Bicycle Physics - Find Constant Velocity

Classical Mechanics Level 3

A bicycle enthusiast decides to design his own bike with one gear ratio. The length of the foot peddle arm is 12 inches from the axis of rotation of the front gear which is 8 inches in diameter. A chain then links the front gear to the rear gear which has a diameter of 4 inches. The bike rear wheel diameter is 29 inches. The rider weighs 200 lbs and is able to exert half of his weight to the peddle while seated. Assuming no acceleration once a constant velocity is achieved, what is his top sustained velocity in mph? Rolling resistance = 0.011. Drag coefficient is 1.0. Density of air is 0.0735 lb-mass/ft^3. The projected area of the bike and biker is 5.5 sft. Assume the bikers weight is evenly distributed between both wheels.

The answer is 9.5.

2 solutions

Steven Chase
Apr 27, 2020

This was a fun problem. I ran the simulation through time for long enough for the speed to reach a steady state. Simulation code is attached. A graph of speed vs. time is also included.

import math

# Basic params

inch = 1.0/12.0

L = 12.0*inch     # foot peddle arm length
Dfg = 8.0*inch    # front gear diameter
Drg = 4.0*inch    # rear gear diameter
Drw = 29.0*inch   # rear wheel diameter
W = 200.0       # rider weight

Crr = 0.011     # rolling resistance coeff
CD = 1.0        # drag coeff
pair = 0.0735   # air density
A = 5.5         # projected area
g = 32.2        # gravity

dt = 10.0**(-5.0)  # time step

###################################################

# Derived params

m = W/g            # system mass
Rrw = Drw / 2.0   # rear wheel radius

Tarm = (W/2.0)*L   # torque to foot peddle arm
Trear = Tarm * (Drg/Dfg)  # torque to rear gear
Frear = Trear/Rrw   # Driving force exerted by rear wheel

###################################################

# Initialize simulation

t = 0.0
count = 0

x = 0.0         # position
xd = 0.0      # velocity
xdd = 0.0    # acceleration

while t <= 20.0:

    x = x + xd*dt               # numerical integration
    xd = xd + xdd*dt

    Froll = Crr*W   # rolling resistance force                       
    FD = 0.5*pair*(xd**2.0)*CD*A  # air drag force

    Fnet = Frear - Froll - FD

    xdd = Fnet/m

    mph = xd*3600.0/5280.0

    t = t + dt
    count = count + 1

    #if count % 1000 == 0:
        #print t,mph

###################################################

print t,mph
#20.0000099994 9.49263852783

@Steven Chase my answer is not matching with your new question's answer of shell potential energy and I have last try also Here is my solution.

A Former Brilliant Member - 1 year, 1 month ago

Here is the math structure I used:

$x = \cos \theta \sin \phi \\ y = \sin \theta \sin \phi \\ z = \cos \phi \\ 0 \leq \theta \leq 2 \pi \\ 0 \leq \phi \leq \pi/2 \\ dS = \sin \phi \, d \theta d \phi \\ dm = \rho \, dS$

Steven Chase - 1 year, 1 month ago

@Steven Chase Okay! But What's the fault in my approach?

A Former Brilliant Member - 1 year, 1 month ago

I see there is a solver now. Is it you?

Steven Chase - 1 year, 1 month ago

@Steven Chase No Sir I can predict that he will Karan Chatrath.

A Former Brilliant Member - 1 year, 1 month ago

@Steven Chase Sir I was making a question. A ball is projected upwards straight. Till the upward motion what is it's expected Potential energy.

A Former Brilliant Member - 1 year, 1 month ago

You mean the average potential energy, over time?