Problem 66 - College Physics

Question

\(\bullet\) 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.

Step-by-Step Solution

Answer

Each rider needs 742.5 W of power.

Step 1: Understand the Concept of Power

Power is the rate at which work is done or energy is transferred. It is calculated using the formula: \( P = F \times v \), where \( P \) is power, \( F \) is the force applied, and \( v \) is the velocity at which the force is applied.

Step 2: Plug Values into Power Formula

Given that the force \( F = 165 \, \mathrm{N} \) and velocity \( v = 9.00 \, \mathrm{m/s} \), we plug these values into the power formula: \[ P = F \times v = 165 \, \mathrm{N} \times 9.00 \, \mathrm{m/s} \].

Step 3: Calculate the Total Power

Perform the multiplication to find the total power required: \[ P = 165 \, \mathrm{N} \times 9.00 \, \mathrm{m/s} = 1485 \, \mathrm{W} \].

Step 4: Distribute Power Equally Between Two Riders

Since the power is to be shared equally between the two riders, divide the total power by 2 to find the power needed per rider: \[ P_{\text{per rider}} = \frac{1485 \, \mathrm{W}}{2} = 742.5 \, \mathrm{W} \].

Key Concepts

Understanding Force

Force is a push or pull upon an object resulting from its interaction with another object. It is a vector quantity, which means it has both magnitude and direction.
In our exercise, the tandem bicycle team needs to overcome a force of 165 Newtons to maintain their speed. This force may arise due to air resistance, friction with the ground, or any other opposing force.
Key points about force:

• Measured in Newtons (N).
• Affects the motion of objects according to Newton's laws.
• Critical in determining how much work or power is needed to move something.

In the context of our problem, understanding the force helps us determine how much effort the cyclists need to apply to maintain their velocity.

Velocity Explained

Velocity is a vector quantity that refers to the rate at which an object changes its position. It combines both the speed and the direction of an object's movement.
For the tandem bicycle, the velocity is given as 9.00 meters per second (m/s). This is the constant speed the bike travels to counteract the force opposing it.
Key ideas about velocity include:

• Velocity tells us how fast something is moving and in which direction.
• Measured in meters per second (m/s) in the SI system.
• Essential for calculating other quantities like momentum and power.

In solving the problem, knowing the velocity is crucial as it directly impacts how much power the riders need to maintain their speed.

The Concept of Work

Work is done when a force causes an object to move. In physics, work is calculated as the product of the force applied to an object and the distance over which that force is applied, aligned with the direction of the force.
However, in the context of power calculation, work is considered in terms of how quickly it is done, which relates to power. Some aspects about work:

• Measured in joules (J) in the SI system.
• Work happens when the force causes movement.
• Work takes into account the direction of force and movement.

Though not explicitly calculated in this problem, understanding work helps connect force, distance, and energy transfer, facilitating problem-solving in physics.

Energy Transfer and Power

Energy transfer occurs when energy moves from one place, object, or form to another. Power measures how fast this transfer happens.
For our tandem bicycle team, the work done by the cyclists through pedaling transfers energy to overcome resistive forces at a rate described by power.Power calculation in our problem involved:

• Multiplying force by velocity, \( P = F \times v \).
• Indicating the rate of energy transfer necessary to maintain motion.
• Expressed in watts (W), with 1 watt equating to 1 joule per second.

Understanding energy transfer and power is fundamental for problems like ours, where determining how much energy is needed over time is crucial for maintaining motion.