Problem 74 - Physics for Scientists and Engineers with Modern Physics

Question

(1II) A bicyclist coasts down a \(6.0^{\circ}\) hill at a steady speed of 4.0 \(\mathrm{m} / \mathrm{s} .\) Assuming a total mass of 75 \(\mathrm{kg}\) (bicycle plus rider), what must be the cyclist's power output to climb the same hill at the same speed?

Step-by-Step Solution

Answer

The cyclist needs approximately 306.64 watts to climb the hill.

Step 1: Identify the Problem

The problem is about finding the power output required for a cyclist to climb a hill at a constant speed of 4.0 m/s. We need to consider gravitational forces due to the hill's incline at 6 degrees, the mass of 75 kg, and the velocity.

Step 2: Calculate Gravitational Force Component

The gravitational force component acting down the incline is calculated using the formula: \[ F_g = mg \sin(\theta) \] where \( m = 75 \text{ kg} \), \( g = 9.8 \text{ m/s}^2 \), and \( \theta = 6^\circ \). Substituting these values: \[ F_g = 75 \times 9.8 \times \sin(6^\circ) \approx 76.66 \text{ N} \]

Step 3: Determine Power Required to Overcome Gravitational Force

Power is calculated as the product of the force and the velocity at which the cyclist climbs the hill, using the formula: \[ P = F \cdot v \] where \( F = 76.66 \text{ N} \) and \( v = 4.0 \text{ m/s} \).Substituting these values: \[ P = 76.66 \times 4.0 \approx 306.64 \text{ W} \]

Step 4: Conclude with the Power Output

Thus, the cyclist needs to exert a power of approximately 306.64 watts to climb the hill at the same steady speed of 4.0 m/s.

Key Concepts

Power Calculation

When dealing with physics problems, calculating power output is an essential skill. Power is the rate at which work is done, and in this context, it refers to the cyclist's effort to maintain a constant speed while climbing an incline. The formula for power is straightforward:

• Power (\( P \)) is calculated as the product of force (\( F \)) and velocity (\( v \)).
• This can be expressed as: \( P = F \cdot v \).

In our example, the cyclist maintains a speed of 4.0 m/s while climbing, which requires overcoming the gravitational force pulling them back down the hill. Knowing how to calculate this allows you to determine how much energy the cyclist needs to sustain effort over time. It's a crucial calculation for understanding performance in cycling and various other activities relying on consistent energy output. With this knowledge, tasks ranging from sports plans to engineering applications become much more manageable.

Gravitational Force

Gravitational force is the natural phenomenon by which all things with mass or energy, including planets, stars, galaxies, and even light, are brought toward one another. In everyday physics problems, gravitational force usually acts downward, toward the center of the Earth. In the context of our problem, it pulls the cyclist down the hill.

• The component of gravitational force (\( F_g \)) that influences movement down an incline can be calculated using the formula: \( F_g = mg \sin(\theta) \).
• Here, \( m \) is mass in kilograms, \( g \) is acceleration due to gravity (9.8 m/s²), and \( \theta \) is the incline angle.

In this example, the gravitational force exerted on the cyclist descending the 6-degree hill is approximately 76.66 N. This force must be overcome to climb at constant velocity, directly involving energy expenditure in terms of mechanical work.

Incline Plane

An inclined plane is a flat surface tilted at a specific angle, as seen in our cycling problem. It's crucial in physics as it introduces component forces, altering the effects of gravity. The angle of the incline affects how objects move along the plane.

• On an incline plane, gravity's force is divided into two components: one parallel and one perpendicular to the surface.
• The parallel component is what the cyclist fights against when climbing, influenced by the sine of the incline angle.

Understanding this allows us to calculate the required power to move the cyclist uphill. Inclines reduce the amount of force necessary to lift objects directly upward, but they increase the distance over which this force must act. Practically, this means a consistent effort is needed over time to maintain speed on an incline, as observed in our example where the cyclist must consistently exert power to climb the hill at 4.0 m/s.