A cyclist intends to cycle up a 7.5 degree hill whose vertical height is 170 m.
Assuming the mass of bicycle plus cyclist is 95 kg, calculate how much work must be done against gravity.
If each complete revolution of the pedals moves the bike 5.6 m along its path, calculate the average force that must be exerted on the pedals tangent to their circular path. Neglect work done by friction and other losses. The pedals turn in a circle of diameter 39 cm.
the work done by gravity is mgh so 95*9.8*170=158270 J
To calculate the work done against gravity, we can use the formula:
Work = Force x Distance x cos(theta)
Where:
Force = mass x gravity
Distance = vertical height
cos(theta) = cosine of the angle of inclination (7.5 degrees)
Given:
Vertical height = 170 m
Mass = 95 kg
Angle of inclination (theta) = 7.5 degrees
Gravity = 9.8 m/s^2
First, let's calculate the force:
Force = mass x gravity
Force = 95 kg x 9.8 m/s^2
Now, calculate the work done against gravity:
Work = Force x Distance x cos(theta)
Work = (95 kg x 9.8 m/s^2) x 170 m x cos(7.5 degrees)
Next, let's calculate the average force exerted on the pedals tangent to their circular path.
Given:
Diameter of the circle = 39 cm
Radius of the circle = (39 cm) / 2 = 0.195 m
Distance traveled in one revolution of the pedals = 5.6 m
The average force exerted on the pedals tangent to their circular path can be calculated using the formula:
Force = Work / Distance
Let's calculate the average force:
Force = Work / Distance
Force = Work / (Distance traveled in one revolution of the pedals)
Now, substitute the values:
Force = Work / (5.6 m)
Now, you can calculate the work done against gravity and the average force exerted on the pedals tangent to their circular path.
To answer this question, we will first calculate the work done against gravity and then determine the average force exerted on the pedals.
1. Work Done Against Gravity:
The work done against gravity can be calculated using the formula:
Work = Force x Distance
In this case, the force is the gravitational force, which can be calculated as the mass of the bicycle plus the cyclist multiplied by the gravitational acceleration (9.8 m/s^2). So, the force is 95 kg x 9.8 m/s^2 = 931 N.
The distance is the vertical height of the hill, which is given as 170 m.
Now we can calculate the work done against gravity:
Work = 931 N x 170 m = 158,170 joules
Therefore, the work done against gravity is 158,170 joules.
2. Average Force Exerted on the Pedals:
To calculate the average force exerted on the pedals, we need to know the distance covered by the bicycle in one revolution of the pedals, which is given as 5.6 m.
The circumference of the circle formed by the pedal rotation can be calculated as the product of the diameter (39 cm) and π (pi). Converting the diameter to meters, we get 0.39 m.
Circumference = π x Diameter = 3.14 x 0.39 m = 1.225 m
Now, we can calculate the average force exerted on the pedals:
Work = Force x Distance
The work done by the cyclist on the pedals is equal to the work done against gravity. So, the work is 158,170 joules.
The distance covered by the bicycle in one revolution of the pedals is 5.6 m.
Therefore, we can rearrange the equation to solve for force:
Force = Work / Distance
Force = 158,170 joules / 5.6 m = 28,250 N
Therefore, the average force that must be exerted on the pedals tangent to their circular path is 28,250 N.