Question

A bicycle and its rider together has a mass of \(75 \mathrm{kg}\). What power output of the rider is required to maintain a constant speed of \(4.0 \mathrm{m} / \mathrm{s}\) (about \(9 \mathrm{mph}\) ) up a \(5.0 \%\) grade (a road that rises \(5.0 \mathrm{m}\) for every \(100 \mathrm{m}\) along the pavement)? Assume that frictional losses of energy are negligible.

Short answer

The rider needs to output 147 watts of power.

Step-by-step solution

Determine the Slope of the Hill

The hill has a grade of 5.0%, meaning it rises 5.0 meters vertically for every 100 meters horizontally. Mathematically, this is expressed as a slope of 0.050 (5/100).

Calculate the Vertical Component of Velocity

The rider's velocity of 4.0 m/s is up the slope, so the vertical component of this velocity is found by multiplying by the sine of the slope angle. Since this is a gentle slope, we approximate \( \sin(\theta) \approx 0.050 \). Thus, the vertical velocity is \( 4.0 \times 0.050 = 0.2 \mathrm{m/s} \).

Compute the Rate of Work Done Against Gravity

The work done against gravity is computed as the product of gravitational force and vertical velocity. The gravitational force is the weight of the rider and bicycle: \( 75 \times 9.8 = 735 \mathrm{N} \). Therefore, the power is \( 735 \times 0.2 = 147 \mathrm{W} \).

Express the Required Power Output

The power output required to maintain this motion and overcome the gravitational component is 147 watts, assuming frictional losses are negligible.

Key concepts

Physics of Motion

Physics of motion is a fascinating subject that deals with how objects move. In our example with the bicycle, we examine linear motion along the inclined road.
Understanding motion requires considering different forces and velocities acting on an object. When a cyclist pedals uphill, they need to exert force to maintain speed against gravity pulling them back down.
The constant speed of 4.0 m/s in this scenario indicates that the applied force by the rider is balanced with the gravitational pull along the slope. This balance is crucial to consistently maintaining motion without acceleration.

Gravitational Force

Gravitational force is the attractive force that acts between any two masses. In the context of our exercise, it specifically refers to the force exerted by the Earth on the bicycle and its rider.
This force acts downwards and is calculated as the product of mass and the acceleration due to gravity, which is approximately 9.8 m/s².

• For the bicycle and rider with a combined mass of 75 kg, the gravitational force amounts to 735 N.
• This force needs to be overcome by the rider's power output to maintain uphill motion.

Inclined Plane

An inclined plane is a flat surface tilted at an angle relative to the horizontal. This simple machine helps reduce the force needed to move objects upwards by spreading the work over a longer distance.
In this problem, the 5.0% grade represents a gentle slope where the road rises 5 meters vertically for every 100 meters of travel.
The incline affects the motion by providing a component of gravitational force parallel to the surface. Effectively, this slope requires extra power for the cyclist to sustain motion as they climb, which is calculated by considering the vertical component of velocity.

Work and Energy

In physics, work is done when a force causes an object to move. The energy required to perform this work is directly related to the power output, both crucial in this cycle problem.
Work done against gravity involves lifting the mass of the bicycle and rider against gravitational pull. The power output, measured in watts, is the rate of doing this work.

• In our exercise, we've calculated a power requirement of 147 W.
• This calculation assumes negligible friction, focusing solely on the energy needed to combat gravity.

By understanding the connection between work, energy, and power, one can grasp how cyclists allocate energy efficiently while tackling uphill climbs.