Problem 55 - University Physics with Modern Physics

Question

A tandem (two-person) bicycle team must overcome a force of 165 N to maintain a speed of 9.00 m/s. Find the power required per rider, assuming that each contributes equally. Express your answer in watts and in horsepower.

Step-by-Step Solution

Answer

The power required per rider is 742.5 W or approximately 0.995 hp.

Step 1: Understand the Formula for Power

Power (P) is defined as the rate at which work is done, which can be calculated using the formula:\[P = F \cdot v\]where F is the force and v is the velocity. We know the team must overcome a force of 165 N and maintain a velocity of 9.00 m/s.

Step 2: Calculate Total Power Required

Substitute the given values into the formula:\[P = 165 \, \text{N} \times 9.00 \, \text{m/s}\]Calculate:\[P = 1485 \, \text{W}\]

Step 3: Calculate Power per Rider

Since each rider contributes equally, divide the total power by the number of riders:\[P_{\text{per rider}} = \frac{1485 \, \text{W}}{2} = 742.5 \, \text{W}\]

Step 4: Convert Power to Horsepower

Use the conversion factor where 1 horsepower is approximately 746 watts:\[P_{\text{hp}} = \frac{742.5 \, \text{W}}{746 \, \text{W/hp}} \approx 0.995 \, \text{hp}\]

Key Concepts

Power Calculation

Calculating power involves determining how quickly work is performed. In physics, power is expressed using the formula:

• \( P = F \cdot v \)
This tells us that power (\(P\)) is the product of force (\(F\)) and velocity (\(v\)).
Force is the push or pull exerted on an object, measured in newtons (\(N\)). Velocity is the speed of the object in a particular direction, measured in \(m/s\) (meters per second).
By multiplying force and velocity, we determine the rate at which work is done. For our tandem bicycle team, a force of 165 \(N\) is needed to maintain a speed of 9.00 \(m/s\), which calculates power by substituting into the formula.

Force and Motion

To fully grasp how power is related to force and motion, it's important to understand what force does. Force causes an object to move or change its motion.

• The greater the force exerted, the greater the potential acceleration of the object.
• In the case of a tandem bicycle, the team must exert a force to counteract resistance from factors like air drag and gravity.

Motion, measured as velocity, is how fast and in which direction the object moves. When force is applied in the direction of motion, work is done, and we can calculate power. For our bicycle team, keeping a consistent velocity of 9.00 m/s requires constant force application to maintain movement and counter usability forces.

Unit Conversion

Unit conversion is vital in physics to move between different measurement systems. Often, results might need to be presented in various units for clarity or to meet specific requirements.

• Watts (\(W\)) is the standard unit of power in the International System of Units.
• Horsepower is another unit often used, especially in vehicular power contexts.

To convert power from watts to horsepower, you use the conversion factor where 1 horsepower equals approximately 746 watts. By dividing the power in watts by this figure, you translate the power output to horsepower, as seen in the exercise where the riders’ power was approximately 0.995 horsepower.

Two-Person Bicycle Team

A tandem bicycle consists of two riders working together. The output from each participant combines to propel the bicycle forward, requiring synchronized pedaling efforts.

• Each rider contributes equally to the work output.
• Power output is split evenly, meaning that the total calculated power must be divided by two.

The efficiency of a tandem team largely depends on their cooperative ability. In this context, each must work at a rate that allows them to overcome the resistance forces together, with each exerting half of the total power necessary to maintain the desired speed. Understanding how these forces and calculations integrate helps riders maximize efficiency and performance. This exercise highlights how working together effectively can help achieve the goal with less effort individually.