Problem 99 - University Physics with Modern Physics

Question

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 \(1983 . )\)

Step-by-Step Solution

Answer

(a) 522.78 W, (b) 374.316 W, (c) 54.63 W.

Step 1: Identify Given Values and Equations

We are given:- Drag coefficient, \(C_d = 1.00\)- Frontal area, \(A = 0.463 \, \text{m}^2\)- Coefficient of rolling friction, \(f_r = 0.0045\)- Rider mass, \(m_r = 50.0 \, \text{kg}\)- Bike mass, \(m_b = 12.0 \, \text{kg}\)- Velocity, \(v = 12.0 \, \text{m/s}\)- Air density, \(\rho \approx 1.225 \, \text{kg/m}^3\)The total force required is the sum of the rolling friction force and the air drag force. We use:1. \( F_{\text{rolling}} = (m_r + m_b) \cdot g \cdot f_r \)2. \( F_{\text{air}} = \frac{1}{2} \cdot C_d \cdot A \cdot \rho \cdot v^2 \)To find the power needed, we use: \( P = F_{\text{total}} \cdot v \) where \( F_{\text{total}} = F_{\text{rolling}} + F_{\text{air}} \).

Step 2: Calculate Forces and Power for Initial Conditions

First, we calculate the force of rolling friction:\[ F_{\text{rolling}} = (50 + 12) \cdot 9.8 \cdot 0.0045 = 2.745 \, \text{N} \]Next, the force of air drag:\[ F_{\text{air}} = \frac{1}{2} \cdot 1.00 \cdot 0.463 \cdot 1.225 \cdot (12)^2 = 40.82 \, \text{N} \]The total force:\[ F_{\text{total}} = 2.745 + 40.82 = 43.565 \, \text{N} \]Finally, the power output by the cyclist:\[ P = 43.565 \cdot 12 = 522.78 \, \text{W} \]

Step 3: Calculate Power Requirement for Racing Conditions

For the racing bike, we change some parameters:- \( C_d = 0.88 \)- \( A = 0.366 \, \text{m}^2 \)- \( f_r = 0.0030 \)- \( m_b = 9.0 \, \text{kg} \)Calculate the rolling friction force:\[ F_{\text{rolling},\text{racing}} = (50 + 9) \cdot 9.8 \cdot 0.003 = 1.743 \, \text{N} \]Calculate the air drag force:\[ F_{\text{air},\text{racing}} = \frac{1}{2} \cdot 0.88 \cdot 0.366 \cdot 1.225 \cdot (12)^2 = 29.45 \, \text{N} \]Total force for racing:\[ F_{\text{total},\text{racing}} = 1.743 + 29.45 = 31.193 \, \text{N} \]Power for racing:\[ P_{\text{racing}} = 31.193 \cdot 12 = 374.316 \, \text{W} \]

Step 4: Calculate Power Requirement at Reduced Speed

For the new condition with speed \( v = 6.0 \, \text{m/s} \) and racing setup:Calculate rolling friction force:\[ F_{\text{rolling},6} = (50 + 9) \cdot 9.8 \cdot 0.003 = 1.743 \, \text{N} \] (same as before)Calculate air drag force:\[ F_{\text{air},6} = \frac{1}{2} \cdot 0.88 \cdot 0.366 \cdot 1.225 \cdot (6)^2 = 7.362 \, \text{N} \]Total force at 6 m/s:\[ F_{\text{total},6} = 1.743 + 7.362 = 9.105 \, \text{N} \]Power at 6 m/s:\[ P_{6} = 9.105 \cdot 6 = 54.63 \, \text{W} \]

Key Concepts

Drag Coefficient

When you're cycling, the drag coefficient plays a pivotal role in determining how much resistance you face from the air. It is a dimensionless number that quantifies the drag or resistance of an object in a fluid environment, such as air or water. The lower the drag coefficient, the more aerodynamic and less resistant an object is.

**Why is this important for cyclists?**

• Reduces the force opposing the cyclist, allowing more efficient movement.
• Critical in racing scenarios, where every fraction of a second counts.

To calculate the force due to air resistance, we use:\[F_{\text{air}} = \frac{1}{2} C_d A \rho v^2\]
Where:

• \( C_d \) is the drag coefficient which we aim to minimize.
• \( A \) is the frontal area, with smaller values leading to less drag.
• \( \rho \) is the air density, typically \( 1.225 \text{kg/m}^3 \) at sea level.
• \( v \) is the velocity of the cyclist.

Cyclists often crouch or use streamlined positions to effectively reduce their drag coefficient and frontal area, making them faster with less effort.

Rolling Friction

Rolling friction refers to the resistance that occurs when an object rolls over a surface. It is typically less than sliding friction, which makes wheels very useful! In cycling, rolling friction impacts how much energy the cyclist needs to maintain a constant speed.

**Factors Affecting Rolling Friction:**

• The material of the tires: Softer tires generally have higher rolling resistance.
• The road surface: Smoother surfaces tend to have lower rolling resistance.
• The weight of the cyclist and bike: More weight increases rolling resistance.

To determine the force due to rolling friction, we use the equation:
\[F_{\text{rolling}} = (m_r + m_b) \cdot g \cdot f_r\]
Where:

• \( m_r \) is the mass of the rider.
• \( m_b \) is the mass of the bicycle.
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{m/s}^2 \).
• \( f_r \) is the coefficient of rolling friction, a number that describes how much force resists the bike's motion.

By optimizing these factors, such as choosing thinner tires or optimizing weight distribution, cyclists can reduce rolling resistance and improve their ride efficiency.

Power Output in Cycling

Power output in cycling is essential for understanding how much energy a cyclist needs to maintain a certain speed. It is the rate at which work is done and is measured in watts (W).

**How is Power Output Calculated?**
To calculate the power output required, we determine the total resisting forces and multiply them by the velocity:\[P = F_{\text{total}} \cdot v\]
Where:

• \( F_{\text{total}} \) is the sum of the forces from rolling friction and air drag:
• \( F_{\text{total}} = F_{\text{rolling}} + F_{\text{air}} \)
• \( v \) is the velocity.

Tips for Cyclists:

• To conserve energy, try to minimize drag and rolling resistance.
• Practicing different cycling positions can help you find more efficient ways to ride.

Understanding power output helps cyclists better manage their energy over long distances and improve their performance strategically.