Problem 74 - Physics for Scientists and Engineers with Modern Physics

Question

(III) 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 power required is approximately 308 W.

Step 1: Convert Hill Angle to Slope

First, we need to determine the slope of the hill using the given angle of inclination. The slope can be calculated using the sine of the angle: \( ext{slope} = \sin(6.0^\circ) \).

Step 2: Calculate Gravitational Force

To find the force of gravity acting parallel to the slope, we use the formula: \( F_{gravity} = m \cdot g \cdot \sin(6.0^\circ) \), where \( m = 75 \text{ kg} \) and \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity.

Step 3: Determine Power Required

Power is calculated by the formula: \( P = F \cdot v \), where \( F \) is the force opposing gravity (which is \( F_{gravity} \), as the cyclist has to counteract this force to climb), and \( v = 4.0 \text{ m/s} \) is the speed. Substituting the values, calculate \( P \).

Key Concepts

Power Output Calculation

Calculating power output is essential in physics, especially when analyzing motion against forces such as gravity. Power measures how quickly work is done or energy is transferred. In the context of a cyclist climbing a hill, power helps us understand the cyclist's energy output required to overcome gravitational pull.

To calculate power output, we use the formula:

• \( P = F \times v \)

where \(P\) is the power in watts, \(F\) is the force in newtons, and \(v\) is the velocity in meters per second. Here, the force \(F\) needed is equal to the gravitational force that acts downward along the slope, and the speed \(v\) is the cyclist’s speed while climbing.

This formula tells us how much work is done by the cyclist per unit of time. More power means more energy is needed to counteract the gravitational pull and maintain a steady speed uphill.

Gravitational Force

Gravitational force is a crucial concept when analyzing movement along an inclined plane. It refers to the force with which the Earth attracts an object towards its center.

For objects on an inclined plane, gravitational force can be split into two components:

• One parallel to the plane (which affects the motion along the plane)
• One perpendicular to the plane (which affects the normal force)

For example, when our cyclist moves uphill, the gravitational force acting parallel to the incline is what resists the upward motion.

The parallel gravitational force \(F_{gravity}\) is calculated using:

• \( F_{gravity} = m \cdot g \cdot \sin(\theta) \)

where \(m\) is the mass, \(g\) is the acceleration due to gravity (approximately \(9.8 \text{ m/s}^2\)), and \(\theta\) is the angle of inclination in degrees.

Understanding gravitational force allows us to determine how much energy or work is needed to overcome its effects, helping to calculate the required power output for uphill movement.

Inclined Plane

An inclined plane is one of the simplest forms of machines that illustrate the physics of motion and forces. It is essentially a flat surface tilted at an angle to the horizontal.

Using an inclined plane allows us to analyze the forces acting on an object moving along the surface. The angle of inclination plays a crucial role as it determines the slope, which affects how much of the gravitational force acts parallel to the surface.

In our cyclist problem, the hill is an inclined plane with an angle of \(6.0^{\circ}\). This angle affects the component of gravitational force that the cyclist must overcome to move upwards.

• A larger angle means more gravitational force acts along the slope, making it harder to climb.
• A smaller angle results in less gravitational resistance, making ascent easier.

Recognizing how inclined planes work is vital for calculating forces like gravitational pull and for understanding how objects move under the influence of various forces.