Cycling. For a touring bicyclist the drag coefficient C(f air =(1)/(2) C A ρv^2) is 1.00, the fr

Cycling. For a touring bicyclist the drag coefficient C(f...

Cycling. For a touring bicyclist the drag coefficient $C\left(f_{\text { air }}=\frac{1}{2} C A \rho v^{2}\right)$ is $1.00,$ the frontal area $A$ is $0.463 \mathrm{m}^{2},$ and the coefficient of rolling friction is $0.0045 .$ The rider has mass 50.0 $\mathrm{kg}$ , and her bike has mass 12.0 $\mathrm{kg}$ (a) To maintain a speed of 12.0 $\mathrm{m} / \mathrm{s}$ (about 27 $\mathrm{mi} / \mathrm{h} )$ on a level road, what must the rider's power output to the rear wheel be? (b) For racing, the same rider uses a different bike with coefficient of rolling friction 0.0030 and mass 9.00 kg. She also crouches down, reducing her drag coefficient to 0.88 and reducing her frontal area to 0.366 $\mathrm{m}^{2} .$ What must her power output to the rear wheel be then to maintain a speed of 12.0 $\mathrm{m} / \mathrm{s} ?$ (c) For the situation in part (b), what power output is required to maintain a speed of 6.0 $\mathrm{m} / \mathrm{s} ?$ Note the great drop in power requirement when the speed is only halved. (For more on aerodynamic speed limitations for a wide variety of human-powered vehicles, "see "The Aerodynamics of Human-Powered Land Vehicles," Scientific American, December $1983 . )$

Key Concepts

Aerodynamic Drag

Aerodynamic drag is the force that opposes an object’s motion through air. It depends on factors such as the speed of the object (typically increasing with the square of the speed), the air density, the object’s frontal area, and its drag coefficient. In scenarios like cycling, overcoming aerodynamic drag is crucial for maintaining higher speeds, which makes understanding this force key to estimating the required power output.

Drag Coefficient

The drag coefficient is a dimensionless number that quantifies the resistance of an object in a fluid environment such as air. It encapsulates the effects of the object’s shape, texture, and flow characteristics. A lower drag coefficient generally indicates a more aerodynamic design, reducing the energy needed to overcome air resistance during motion.

Frontal Area

Frontal area represents the cross-sectional area of an object that faces the fluid flow—in this context, the air. A larger frontal area results in a higher aerodynamic drag because more air is disturbed as the object moves, leading to greater resistive forces that must be countered by additional power.

Rolling Resistance

Rolling resistance, often expressed using the coefficient of rolling friction, is the frictional force that resists the motion of a rolling object on a surface. It is typically proportional to the normal force (which depends on the mass of the object and gravity) and is a significant factor in determining the overall power required, particularly at lower speeds or on level roads.

Power Requirement

The power requirement is the rate at which work must be done to overcome resistive forces like aerodynamic drag and rolling resistance. Since both the force overcome and the distance traveled per unit of time (i.e., speed) factor into work, the power needed scales with the product of these forces and the velocity, often showing nonlinear behavior, especially with respect to speed.

Velocity Dependence

The forces acting on a moving cyclist, especially aerodynamic drag, depend strongly on the speed. While rolling resistance is relatively constant with speed, aerodynamic drag increases with the square of the velocity, causing the power required to overcome it to increase roughly with the cube of the speed. This sensitivity to velocity is crucial in understanding why small changes in speed can result in large differences in required power output.

Transcript

00:01 For this problem on the topic of work in kinetic energy, we are given the drag coefficient for a touring cyclist, the frontal area, as well as the coefficient of rolling friction.

00:14 If we know the mass of the rider and the mass of the bike, we want to calculate the rider's power output to the rear wheel to maintain a speed of 12 meters per second.

00:27 Then we are told that for racing, the same rider will use a different bike with a different coefficient of rolling friction and mass.

00:35 The rider also crouches down to reduce the drag coefficient to another given value and a lower frontal area.

00:41 We now want to calculate the power output again to the real wheel that must be given to maintain the same speed of 12 meters per second.

00:50 And lastly, for this situation in part b, we want to know the power output that is required to maintain a speed that is half the speed before, now six meters per second.

01:03 So first, firstly, in part a, to calculate the power p, we know this is the total force f total times the velocity v.

01:17 But f total, remember, is equal to the force due to rolling friction plus the force due to air resistance.

01:31 So we know the force due to air resistance is given by the equation a half times c times the frontal area a times row v squared.

01:48 So this is a half times the coefficient, the dry coefficient 1 times the frontal area given as 0 .463 meters square.

02:07 Times the density of air 1 .2 kg per cubic meter times the constant speed of 12 meters per second all squared.

02:24 This gives us the force of air resistance to be 40 newtons.

02:32 The force duty rolling friction f roll is equal to the coefficient of rolling friction mu r times the normal force n which is equal to new r times the weight of the cyclone bike w and this is equal to 0 .0045 into 400 and 90 newtons plus 118 neutins which gives us a rolling force of 2 .74 now that we have both the forces, we can calculate the power.

03:20 So therefore the power is equal to the rolling force plus the force due to air resistance times the velocity v.

03:34 So that's 2 .74 newtons plus 40 newtons multiplied by the constant speed of 12 meters...

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