Problem 53 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

Each rider requires 742.5 watts or 0.996 horsepower.

Step 1: Understanding Power and Force

Power is calculated as the work done per unit of time. Since the bicycle is moving at a constant speed, the power can also be calculated using the formula: \[ P = F \times v \] where \( P \) is power, \( F \) is the force, and \( v \) is the velocity. In this problem, \( F = 165 \, \mathrm{N} \) and \( v = 9.00 \, \mathrm{m/s} \).

Step 2: Calculating Total Power

We substitute the given values into the power formula:\[ P = 165 \, \mathrm{N} \times 9.00 \, \mathrm{m/s} = 1485 \, \mathrm{W} \] Thus, the total power required to maintain the speed is 1485 watts.

Step 3: Determining Power per Rider

Since each rider contributes equally, the power required per rider is half of the total power calculated:\[ P_{\text{per rider}} = \frac{1485 \, \mathrm{W}}{2} = 742.5 \, \mathrm{W} \]

Step 4: Converting Watts to Horsepower

To convert the power from watts to horsepower, we use the conversion factor: \( 1 \, \text{horsepower} = 745.7 \, \mathrm{W} \).So, \[ P_{\text{per rider in hp}} = \frac{742.5 \, \mathrm{W}}{745.7} = 0.996 \, \text{horsepower} \]

Key Concepts

Power Calculation

Power calculation is fundamental in physics when determining how much work is done over a period of time. When bicyclists pedal, they exert a force to overcome friction and air resistance. Power can thus be calculated by using the formula:
\[ P = F \times v \]
where \( P \) represents power, \( F \) is the force applied, and \( v \) is the velocity at which the object moves. This formula is particularly useful in scenarios like the tandem bicycle example provided.
The force is given as 165 N, and the velocity is 9.00 m/s, so when multiplying these two numbers, we find the total power needed to maintain the bicycle's speed. Always remember, using this formula requires understanding that the direction of force and motion are the same.

Force and Velocity

Force and velocity are two integral concepts in understanding motion. Force, measured in Newtons (N), is any interaction that changes the motion of an object.
For the tandem bicycle, maintaining a constant velocity of 9.00 \( \mathrm{m/s} \) requires overcoming a steady force of 165 \( \mathrm{N} \).
Velocity, on the other hand, is the speed of an object in a particular direction. It tells us not just how fast something is moving, but also where it is headed. The constant velocity means the bicycle's speed does not change, indicating that the power used is solely to counteract resistive forces, such as friction and air resistance.
In practical scenarios, analyzing both force and velocity helps in understanding the energy dynamics involved in movement.

Unit Conversion

Unit conversion is a crucial skill in solving physics problems, as it allows for the translation between different measures used in specific contexts.
In the tandem bicycle example, after determining the power required per rider to be 742.5 watts, it's insightful to convert this power into other units like horsepower for a clearer understanding, especially in regions where horsepower is a more familiar unit.
To convert watts to horsepower, the conversion factor used is:

• 1 horsepower = 745.7 watts

Thus, converting 742.5 watts gives approximately 0.996 horsepower per rider.
Such conversions are helpful for making different numerical values intuitive to different audiences and for ensuring the compatibility of numbers across varied industrial and scientific standards.