Question

A bicyclist coasts down a \(7.0^{\circ}\) slope at a steady speed of \(5.0 \mathrm{~m} / \mathrm{s}\). Assuming a total mass of \(75 \mathrm{~kg}\) (bicycle plus rider), what must the cyclist's power output be to pedal up the same slope at the same speed?

Short answer

Answer: The cyclist's power output is approximately 447.85 watts.

Step-by-step solution

Calculate the force of gravity

To find the force of gravity acting on the cyclist as they pedal up the hill, we can use the formula: $$ F_g = mg \sin \theta $$ Given the mass \(m = 75 \; kg\), the acceleration due to gravity \(g = 9.81 \; \frac{m}{s^2}\), and the angle of the slope \(\theta = 7.0º\), we first need to convert the angle to radians: $$ \theta_{rad} = \frac{\pi}{180}\theta = \frac{\pi}{180}(7.0) = 0.122 \; rad $$ Now, we can calculate \(F_g\): $$ F_g = (75 \cdot 9.81) \sin (0.122) = 735.75 \cdot 0.1218 \approx 89.57 \; N $$

Calculate the power output

Now that we have the force of gravity acting on the cyclist, we can find the power output using the formula for power: $$ P = F_g \times v $$ Given the cyclist's speed is \(5.0 \; \frac{m}{s}\), we can find the power output as follows: $$ P = 89.57 \times 5.0 = 447.85 \; W $$ Therefore, the cyclist's power output must be approximately 447.85 watts to pedal up the slope at the same speed.

Key concepts

Force of Gravity Calculation

Understanding how the force of gravity affects an object in motion is crucial in physics, especially when that object is on an incline, like our cyclist. Gravity's force is a constant pull towards the center of the Earth and is determined by an object's mass and the acceleration due to gravity, which is approximately 9.81 meters per second squared on Earth's surface. However, not all this force acts to pull the cyclist down the slope; only a component of it does.

The component of the gravitational force acting parallel to the incline is found using the formula: \[ F_g = mg \sin(\theta) \] Here, \(mg\) represents the full weight of the cyclist and their bike, and \(\sin(\theta)\) gives us the ratio of the gravitational force acting down the slope. The angle of the slope, \(\theta\), must be in radians for the calculations to work correctly. In our problem, this calculation helps us understand the challenge the cyclist must overcome to ride up the hill. To make sense of this mathematically, the steeper the slope (a larger value of \(\theta\)), the greater the force the cyclist will need to work against. It's important to remember individual components when analyzing forces in real-life scenarios like this.

Power Output Calculation

After understanding the force acting on the cyclist due to gravity, the next step is to quantify the effort needed to conquer this force, known as power output. Power in physics is the rate of doing work, or in simpler terms, how much effort is needed per unit of time. In the context of our exercise, it represents how hard the cyclist must pedal to maintain a constant speed uphill.

The power output can be calculated with the formula: \[ P = F_g \times v \] where \(P\) stands for power, \(F_g\) is the force of gravity we calculated earlier, and \(v\) is the cyclist's velocity. The product of force and velocity gives us the power in watts. It's essential to note that only the force component in the direction of motion (up the incline) must be used when calculating power output. This calculation is not only important for understanding physics problems but also has practical applications, such as determining the capabilities of electric motors or the fitness level of an athlete.

Motion on an Incline

Motion on an incline introduces complexities because it combines both vertical and horizontal movements. For any object moving on an incline, two main forces come into play: the gravitational force pulling it down the slope and the force exerted to move it up or down that slope. The angle of the incline significantly affects the motion because it changes the effective weight of the object, meaning the force it feels due to gravity, which is directly related to how much it will accelerate.

For objects moving at a steady speed, like our cyclist coasting downhill, there's a balance between the force of gravity and the forces of friction and air resistance. Conversely, when going uphill, the cyclist must exert a force greater than the gravitational component to move up at a constant speed. Motion on an incline is a common real-world problem and encountering it in physics is a great way to understand key concepts such as components of forces, work, energy, and power.