Problem 24 - Physics for Scientists and Engineers with Modern Physics

Question

(1I) A cyclist intends to cycle up a \(9.50^{\circ}\) hill whose vertical height is 125 \(\mathrm{m}\) . The pedals turn in a circle of diameter 36.0 \(\mathrm{cm} .\) Assuming the mass of bicycle plus person is 75.0 \(\mathrm{kg}\) , (a) calculate how much work must be done against gravity. (b) If each complete revolution of the pedals moves the bike 5.10 \(\mathrm{m}\) 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.

Step-by-Step Solution

Answer

(a) 91837.5 J, (b) 1092.3 N

Step 1: Calculate Total Work Against Gravity

To find the work done against gravity, we use the formula for gravitational work: \[ \text{Work} = m \cdot g \cdot h \]where\( m = 75.0 \, \text{kg} \) (mass of the cyclist and bike),\( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity),and \( h = 125.0 \, \text{m} \) (vertical height).Substitute these values into the formula:\[ \text{Work} = 75.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 125.0 \, \text{m} = 91837.5 \, \text{J} \]

Step 2: Calculate Distance Pedaled

To determine the vertical distance converted into work per pedal revolution, calculate how many revolutions are needed to climb the hill:First, find the path distance using trigonometry:\[ \sin(9.50^\circ) = \frac{h}{\text{path distance}} \]We rearrange to find 'path distance':\[ \text{path distance} = \frac{h}{\sin(9.50^\circ)} = \frac{125 \, \text{m}}{\sin(9.50^\circ)} \approx 756.8 \, \text{m} \]

Step 3: Calculate Number of Pedal Revolutions

Now, compute the number of pedal revolutions required:\[ \text{Revolutions} = \frac{\text{path distance}}{\text{distance per revolution}} = \frac{756.8 \, \text{m}}{5.10 \, \text{m/rev}} \approx 148.2 \, \text{revolutions} \]

Step 4: Calculate Average Force Exerted on Pedals

We will calculate the average force exerted tangent to the pedal’s path using:\[ \text{Work} = \text{Force} \times \text{Distance} \]Substituting work (91837.5 J) and distance covered over 148.2 revolutions (the circumference times the number of revolutions, \(148.2 \times \pi \times 0.18 \, \text{m} \)): \[ \text{Distance} = 148.2 \times \pi \times 0.18 = 84.07 \, \text{m} \]\[ \text{Force} = \frac{91837.5}{84.07} \approx 1092.3 \, \text{N} \]

Key Concepts

Gravitational Work

Gravitational work is the energy needed to move an object against the gravitational force. In the given exercise, we calculate this work as the cyclist ascends a hill of a certain vertical height. The formula for gravitational work is:
\[ \text{Work} = m \cdot g \cdot h \]
where:

• \( m \) is mass (75 kg for the cyclist and bike combined).
• \( g \) is the acceleration due to gravity (approx. 9.81 m/s²).
• \( h \) is the vertical height to be climbed (125 meters).

Plugging in these values, we get:
\[ \text{Work} = 75 \times 9.81 \times 125 = 91837.5 \, \text{Joules} \]
This result means the cyclist must do about 91,838 Joules of work against gravity to reach the top of the hill.
This scenario exemplifies how energy is conserved and transformed, demonstrating an essential principle of physics.

Force Calculation

Calculating the force exerted on the pedals is crucial for determining the cyclist's effort in climbing the hill. To find this force, we need to know the distance the pedals travel and the work done.
First, calculate the work done per pedal revolution:

• The path distance the bike travels is determined using trigonometry, resulting in 756.8 meters for the entire hill climb.
• With 5.10 meters per pedal revolution, divide the total path distance by this to find the total revolutions: 148.2 revolutions.

Next, calculate the force:
Use the relationship: \( \text{Work} = \text{Force} \times \text{Distance} \).
The total distance the pedals travel is the circumference of their path (\( \pi \times 0.18 \, \text{m} \)) times the number of revolutions:

• \( 148.2 \times \pi \times 0.18 \approx 84.07 \, \text{m} \).

Finally, solve for force:
\[ \text{Force} = \frac{91837.5}{84.07} \approx 1092.3 \, \text{Newtons} \]
This result tells us the average force applied to the pedals is approximately 1092 Newtons, emphasizing the physical effort needed in cycling uphill.

Circular Motion

When a cyclist pedals, the motion is circular, specifically when looking at how the pedal rotates around its axis. Understanding this motion helps in calculating the force exerted and distance traveled by the pedals.
In this exercise, the pedals form a circle with a diameter of 36 cm (or 0.36 meters). The circular path has some key characteristics:

• The radius (half the diameter) is 0.18 meters.
• The circumference, which is the total circular distance a pedal covers in one spin, is \( \pi \times 0.36 \).

Given the rotational motion of the pedals, each complete revolution moves the bike 5.10 meters along its path. Therefore, understanding this circular motion aids in determining how much work is done and what distance each pedal stroke covers.
The blend of linear and circular motion concepts is a practical illustration of kinetic energy at play, showing how circular motions translate to linear travel, an important aspect of mechanical physics and everyday cycling.