91Ó°ÊÓ

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 \(\mathrm{kg}\) . 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 \((\mathrm{b}),\) 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.

Step by step solution

01

Calculate Total Force Required

To maintain a constant speed, the rider needs to overcome two main forces: rolling resistance and air drag. The rolling resistance force, \(f_{\text{rolling}}\), is given by:\[f_{\text{rolling}} = \text{coefficient of rolling friction} \times \text{total weight}\]where the total weight is the sum of the rider's and bike's mass multiplied by gravity (\(g = 9.8 \, \text{m/s}^2\)).The drag force, \(f_{\text{air}}\), is given by:\[f_{\text{air}} = \frac{1}{2} \cdot C \cdot A \cdot \rho \cdot v^2\]where \(C\) is drag coefficient, \(A\) is frontal area, \(\rho\) is air density (which we assume as \(1.225 \, \text{kg/m}^3\)), and \(v\) is velocity.

02

Solve (a) - Calculate Power Output for Part (a)

First, calculate the rolling resistance for part (a):\[f_{\text{rolling}} = 0.0045 \times (50.0 + 12.0) \times 9.8\]Next, calculate the air drag force:\[f_{\text{air}} = \frac{1}{2} \times 1.00 \times 0.463 \times 1.225 \times (12.0)^2\]Then, find total force:\[F_{\text{total}} = f_{\text{rolling}} + f_{\text{air}}\]Finally, calculate power:\[P = F_{\text{total}} \times v = F_{\text{total}} \times 12.0\]

03

Solve (b) - Calculate Power Output for Racing Bike

Repeat the calculations from Step 2 using the racing bike data:Rolling resistance for part (b):\[f_{\text{rolling}} = 0.0030 \times (50.0 + 9.0) \times 9.8\]Air drag with new values:\[f_{\text{air}} = \frac{1}{2} \times 0.88 \times 0.366 \times 1.225 \times (12.0)^2\]Total force:\[F_{\text{total}} = f_{\text{rolling}} + f_{\text{air}}\]Power:\[P = F_{\text{total}} \times 12.0\]

04

Solve (c) - Calculate Power Output for Slower Speed

Use values from part (b) but change speed to 6 m/s:Air drag:\[f_{\text{air}} = \frac{1}{2} \times 0.88 \times 0.366 \times 1.225 \times (6.0)^2\]Rolling resistance remains the same as in part (b).Total force:\[F_{\text{total}} = f_{\text{rolling}} + f_{\text{air}}\]Power:\[P = F_{\text{total}} \times 6.0\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rolling Resistance

Rolling resistance is one of the key factors that cyclists must overcome to maintain a steady speed. It is the resistance that occurs between the bicycle tires and the road surface.
This force depends on a few critical elements:

• The coefficient of rolling friction, which is influenced by tire material and surface.
• The combined weight of the cyclist and the bicycle.
• The gravitational force acting on the system.

To calculate the rolling resistance force, the formula is:\[f_{\text{rolling}} = \text{coefficient of rolling friction} \times \text{total weight} \times g\]Here, the total weight is the sum of the rider's and bike’s masses, and \(g\) represents gravity (approximately \(9.8 \, \text{m/s}^2\)).
For example, if the cyclist and bike together weigh 62 kg and the coefficient of rolling friction is 0.0045, the rolling resistance will exert a constant force that the cyclist needs to overcome to move at a constant speed.

Drag Force Calculation

Another significant force that impacts a cyclist is the air drag or aerodynamic drag. This force is more pronounced at higher speeds and depends on factors such as the cyclist's speed, the density of the air, the drag coefficient, and the frontal area of the cyclist and the bicycle.The formula to find the drag force is:\[f_{\text{air}} = \frac{1}{2} \times C \times A \times \rho \times v^2\]

• \(C\) is the drag coefficient, typically determined by the shape and posture of the rider.
• \(A\) is the frontal area, how much of the rider is "hitting" the wind directly.
• \(\rho\) is the air density, standardly \(1.225 \, \text{kg/m}^3\).
• \(v\) is the velocity or speed of the cyclist.

A lower drag coefficient and reduced frontal area greatly reduce the drag force, allowing for more efficient cycling, especially in racing scenarios where every bit of reduced resistance helps.

Power Output in Cycling

Power output in cycling is a critical measurement of how much energy a cyclist needs to exert to overcome resistance forces and maintain a particular speed.
Power is calculated as the product of the total resistance force and the cyclist’s velocity:\[P = F_{\text{total}} \times v\]The total force \(F_{\text{total}}\) is the sum of rolling resistance (\(f_{\text{rolling}}\)) and air drag (\(f_{\text{air}}\)).Factors affecting power output include:

• Velocity: Higher speeds dramatically increase the power required, primarily due to the quadratic relationship in the drag formula.
• Equipment and posture: Lightweight bikes and aerodynamic positions can lower rolling resistance and drag.
• Terrain: Uphill rides increase the gravitational component of the force to overcome.

Understanding where energy is spent helps riders optimize their efforts and improve performance by minimizing unnecessary resistance.