Problem 82 - Physics for Scientists and Engineers with Modern Physics

Question

A bicyclist of mass \(75 \mathrm{~kg}\) (including the bicycle) can coast down a \(4.0^{\circ}\) hill at a steady speed of \(12 \mathrm{~km} / \mathrm{h}\). Pumping hard, the cyclist can descend the hill at a speed of \(32 \mathrm{~km} / \mathrm{h}\). Using the same power, at what speed can the cyclist climb the same hill? Assume the force of friction is proportional to the square of the speed \(v\), that is, \(F_{\mathrm{fr}}=b v^{2}\), where \(b\) is a constant.

Step-by-Step Solution

Answer

The cyclist can climb the hill at approximately 3.57 m/s.

Step 1: Convert Speeds to m/s

First, convert the given speeds from km/h to m/s.1. For coasting speed: \[12 \text{ km/h} = \frac{12 \times 1000}{3600} = 3.33 \text{ m/s}\] 2. For descending speed with pumping hard: \[32 \text{ km/h} = \frac{32 \times 1000}{3600} = 8.89 \text{ m/s} \]

Step 2: Determine Forces and Power

For coasting down the hill:1. At steady speed, gravity's component down the hill balances friction: \[ mg \sin(4^\circ) = bv^2 \] Substituting values, \[ 75 \times 9.8 \times \sin(4^\circ) = b(3.33)^2 \]For descending the hill with additional speed:1. Effective power used is the difference in gravitational force and friction at 32 km/h.2. Power \(P = F_{\text{net}} \times v\) can be calculated.

Step 3: Calculate Power Used While Descending

From the balance of forces while coasting, use the known coasting speed to find the frictional constant \(b\), and then calculate the power:1. Calculate \(b\):\[ b = \frac{75 \times 9.8 \times \sin(4^\circ)}{(3.33)^2} \]2. Find effective power \(P\) during descent at \(8.89 \text{ m/s}\):\[ P = \left( mg \sin(4^\circ) - b(8.89)^2 \right) \times 8.89 \]

Step 4: Determine Speed for Climbing the Hill

Use the calculated power to determine the speed \(v\) at which cyclist can climb:1. Frictional force while climbing will be: \[ bv^2 \]2. Power \(P\) is also equal to the climbing friction force times speed: \[ P = (mg \sin(4^\circ) + bv^2) \cdot v \]3. Solve for \(v\) in this equation using the previously determined power \(P\) and friction constant \(b\).

Step 5: Solve for Climbing Speed

Solve the final power equation:\[ P = (75 \times 9.8 \times \sin(4^\circ) + bv^2) \times v \]Substitute the known power and solve for \(v\).This will generally involve solving a cubic equation, so typically numerical or iterative methods, or graphing tools may be necessary.

Key Concepts

Friction Force

In physics, friction is a force that opposes the motion of objects or surfaces. Friction can be categorized into two types: static and kinetic. In this exercise, we focus on kinetic friction which acts on moving objects like a descending bicyclist.
The friction force can be mathematically described as:

• Proportional to the normal force, or contact force between surfaces.
• A function of the surfaces' coefficient of friction.

In this particular problem, the frictional force is simplified to depend on the square of the speed (\( v \)) as given by the formula: \( F_{\mathrm{fr}} = b v^2 \).
This relationship indicates that as the cycling speed increases, the frictional resistance increases significantly. Your job is essentially figuring out how much work or power a cyclist needs to overcome this increasing resistance under different circumstances, such as ascending or descending an inclined hill.

Power Calculation

Power is the rate at which work is done or energy is transferred over time. In this scenario, power is used to describe how the cyclist applies energy to overcome friction and to maintain or achieve a certain speed on the hill.
The formula for power (\( P \)) is given by \( P = F_{\text{net}} \times v \), where \( F_{\text{net}} \) is the net force after accounting for forces such as gravity and friction.
To determine the power utilized by or against the cyclist, it's necessary to:

• Calculate the gravitational component working along the incline.
• Determine frictional forces opposing motion.
• Measure the difference when increasing speed due to additional power while descending.

In this particular exercise, the key is finding the constant power which allows the cyclist to both descend and ascend a given hill efficiently.

Newton's Laws of Motion

Newton's Laws of Motion give a fundamental framework for understanding how objects behave under different forces. These laws are essential in this physics problem.
Firstly, Newton's First Law, or the Law of Inertia, states that an object in motion stays in motion unless acted upon by an external force. Gravity, friction, and additional cyclist input affect the system's inertia as it goes down or uphill.
Newton's Second Law is most relevant here, expressed as \( F = ma \), where the net force acting on the cyclist equals mass (\( m \)) times acceleration (\( a \)). Since friction balances the forces when coasting, the net force applied when changing speeds can calculate power or climbing capabilities.
Finally, while Newton's Third Law—every action has an equal and opposite reaction—is related, it's less central to solving for power or speed as the bicycle's motor actions adapt to opposing forces on the inclined plane.

Inclined Plane

An inclined plane is a flat supporting surface tilted at an angle, with one end higher than the other, like the hill described in the problem. It’s a classic example of simple machines used to study dynamics.
Studying a bicycle on an incline involves understanding both the gravitational force pulling it down and forces resisting motion like friction and air resistance.
Key concepts include:

• Gravitational force parallel to the incline: \( mg \sin(\theta) \)
• Normal force, perpendicular to the incline: \( mg \cos(\theta) \)

On an incline, friction is tied to both the normal force and any additional forces applied (such as the cyclist pedaling harder). So, understanding the relationship between gravity, friction, and inclined planes is critical for solving problems that involve motion on a slope, like ascending or descending efficiently.