A bicyclist maintains a constant speed of 4.00 m/s up a hill that is inclined at 10.0
∘
with the horizontal. Calculate the work done by the person and the work done by gravity if the bicycle moves a distance of 18.0 m up the hill. The combined mass of the rider and the bike is 82.0 kg.
To calculate the work done by the person and the work done by gravity, we need to use the formula for work:
Work = Force × Distance × cos(θ)
Where:
- Work is the amount of energy transferred by a force when an object is moved a certain distance.
- Force is the magnitude of the force applied on the object.
- Distance is the magnitude of the displacement of the object.
- θ (theta) is the angle between the direction of the force and the direction of displacement.
Let's calculate each term separately.
1. Work done by the person:
The force exerted by the person on the bike can be calculated using Newton's second law:
Force = Mass × Acceleration
Given that the combined mass of the rider and the bike is 82.0 kg and the acceleration is zero (since the speed is constant), the force exerted is:
Force = 82.0 kg × 0 m/s² = 0 N
Since the force and the displacement are in the same direction (up the hill) and the angle (θ) is 0 degrees, the cos(θ) term is 1:
cos(0°) = 1
Therefore, the work done by the person is:
Work = Force × Distance × cos(θ)
Work = 0 N × 18.0 m × 1
Work = 0 J (Joules)
2. Work done by gravity:
The force of gravity can be calculated using the formula:
Force = Mass × Gravity
Where the mass is 82.0 kg and the acceleration due to gravity is approximately 9.8 m/s²:
Force = 82.0 kg × 9.8 m/s² = 803.6 N
Since the force and the displacement are in opposite directions (up the hill and down the hill respectively), the angle (θ) is 180 degrees, and cos(θ) is -1:
cos(180°) = -1
The distance up the hill is given as 18.0 m. However, since the work done by gravity is the work against gravity's force, we need to consider a negative displacement:
Distance = -18.0 m
Therefore, the work done by gravity is:
Work = Force × Distance × cos(θ)
Work = 803.6 N × -18.0 m × -1
Work = 14,465.6 J (Joules)
So, the work done by the person is 0 J, and the work done by gravity is 14,465.6 J.