Problem 24 - Physics for Scientists and Engineers with Modern Physics

Question

(II) 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) Work done against gravity is 91,875 J. (b) Average force on the pedals is approximately 201.5 N.

Step 1: Determine Work Done Against Gravity

To find out how much work needs to be done against gravity, we use the formula for work: \[ W = mgh \] where \( m \) is the mass of the cyclist and bicycle combined, \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)), and \( h \) is the vertical height of the hill. Substitute the values:\[ W = 75.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 125 \, \text{m} \] Calculate the work: \[ W = 91,875 \, \text{Joules} \] So, the work done against gravity is \( 91,875 \, \text{J} \).

Step 2: Calculate the Work Done Per Revolution

Next, we need to understand how much distance is covered per pedal revolution. We know that each revolution covers \( 5.10 \, \text{m} \) along the hill path. To maintain the cyclist's constant uphill motion, we need to apply a force over this distance, equal to the work against gravity per revolution.

Step 3: Calculate the Average Force Exerted on the Pedals

The work done against gravity must equal the force exerted on the pedals times the distance moved along the path per revolution. We use the relation:\[ W = F \times d \] Rearranging for the force, we get:\[ F = \frac{W}{d} \] Substituting the values (we use the work for a full revolution and the distance covered):\[ F = \frac{91,875}{5.10} \approx 18,015.7 \, \text{Newtons} \] Thus, the average force that must be exerted on the pedals is approximately \( 18,015.7 \, \text{N} \).

Step 4: Refine and Calculate Force on Pedals

Since the force must be tangential to the circle's path, use the pedal's circular path for additional accuracy. The diameter of the pedal circle is \( 36.0 \, \text{cm} \), so the radius is \( 0.18 \, \text{m} \). Therefore, the circumference (complete revolution length) is:\[ C = 2\pi \times 0.18 \, \text{m} = 0.36\pi \, \text{m} \] The average force must be adequate to produce the equivalent work over this circle length. Adjust the force calculation:\[ F_{\text{avg}} = 18,015.7 \, \text{N} \times \frac{0.36\pi}{5.10} \approx 201.5 \, \text{Newtons} \]Thus, the actual average force to be exerted on each pedal is approximately 201.5 N.

Key Concepts

Work against gravity

When you're cycling up a hill, you're working against the force of gravity. This work can be quantified using the formula \( W = mgh \), where:

• \( m \) is the mass of the cyclist and bicycle combined, in kilograms.
• \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \text{m/s}^2 \).
• \( h \) is the vertical height of the hill, in meters.

Using this formula helps us determine how much energy, or work, is required to move upwards against gravity. In the exercise, the vertical height \( h \) is given as \( 125 \, \text{m} \) and the mass \( m \) is \( 75.0 \, \text{kg} \). Substituting these values, we find that \( W = 75.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 125 \, \text{m} = 91,875 \, \text{J} \) (Joules). This figure represents the total work needed to overcome gravitational force while climbing the hill.

Force calculation

Force plays a critical role in moving an object, particularly in cycling uphill. The key principle here is that work done is equal to the force applied over a distance. Mathematically, this is expressed as \( W = F \times d \), where:

• \( W \) is the work done, measured in Joules.
• \( F \) is the force applied, in Newtons.
• \( d \) is the distance over which the force is applied, in meters.

In the context of our problem, each revolution of the pedal moves the bike \( 5.10 \, \text{m} \) along its path. To find the force exerted on the pedal, we rearrange the formula to \( F = \frac{W}{d} \). Given \( W = 91,875 \, \text{J} \) and \( d = 5.10 \, \text{m} \), the force \( F \approx 18,015.7 \, \text{N} \). This value is adjusted for the actual pedal path, considering the pedal's circular motion, resulting in an average force of about \( 201.5 \, \text{N} \) being exerted per pedal turn.

Energy conversion

Cycling uphill involves converting energy between different forms. Primarily, your body converts chemical energy from food into kinetic energy to move the bike. However, moving up an incline involves converting some of this kinetic energy into gravitational potential energy.Gravitational potential energy \( (U) \) is defined as \( U = mgh \), which is the energy stored by virtue of the bike and rider being elevated against gravity. As the cyclist climbs the hill, the work done transfers energy into this form, emphasizing the law of energy conservation:

• Energy is not lost; it is transformed from one type to another.
• The objective is to use energy efficiently, converting enough of it to climb the hill while maintaining forward motion.

In this exercise, the total work required against gravity is \( 91,875 \, \text{J} \), which corresponds to the energy converted and stored as gravitational potential energy as the cyclist reaches the peak of the hill.