Problem 4 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A \(56.5-\mathrm{kg}\) hiker starts at an elevation of \(1270 \mathrm{~m}\) and climbs to the top of a 2660-m peak. (a) What is the hiker's change in potential energy? (b) What is the minimum work required of the hiker? \((c)\) Can the actual work done be greater than this? Explain.

Step-by-Step Solution

Answer

(a) 771,621.15 J; (b) 771,621.15 J; (c) Yes, due to inefficiencies.

Step 1: Determine the Change in Elevation

Identify the initial and final elevations from the problem statement. The hiker starts at 1270 m and climbs to 2660 m. Calculate the change in elevation: \( \Delta h = 2660 \text{ m} - 1270 \text{ m} = 1390 \text{ m} \).

Step 2: Calculate the Change in Potential Energy

Use the formula for gravitational potential energy: \( \Delta PE = m \cdot g \cdot \Delta h \), where \( m = 56.5 \text{ kg} \) is the mass of the hiker, \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity, and \( \Delta h = 1390 \text{ m} \). Calculate: \[ \Delta PE = 56.5 \cdot 9.81 \cdot 1390 = 771,621.15 \text{ Joules} \]

Step 3: Determine the Minimum Work Required

The minimum work required to lift the hiker to the peak is equal to the change in potential energy. Therefore, the minimum work required is 771,621.15 Joules.

Step 4: Discuss the Actual Work Done

Consider factors that might cause the actual work to exceed the minimum. The actual work done can be greater due to inefficiencies such as friction, air resistance, and the hiker's own effort inefficiency, meaning the actual energetic expenditure could exceed the ideal case of only overcoming gravitational force.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is a type of energy associated with the position of an object in a gravitational field. It depends on three main factors:

• The mass of the object ( \( m \) ).
• The height of the object above a reference point ( \( h \) ).
• The gravitational acceleration ( \( g = 9.81 ext{ m/s}^2 \) on Earth).

Gravitational potential energy (GPE) can be calculated using the formula: \[ PE = m \cdot g \cdot h \]This means that the energy increases when either the mass or height of the object increases.
So for the hiker, who weighs 56.5 kg and ascends to a higher altitude, there's a visible increase in potential energy because they've been elevated higher in Earth's gravity field. When solving problems involving gravitational potential energy, always ensure to accurately determine height change, as it's this shift that predominantly affects potential energy changes.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics, linking the idea of work done on an object to changes in its energy state, such as potential and kinetic energy. According to this principle, the work done on an object is equal to the change in its energy:\[ W = \Delta KE + \Delta PE \]For our hiking scenario, the principle helps calculate the necessary work to increase the hiker's gravitational potential energy. All the energy exerted in hiking goes into overcoming gravity and inefficient factors like energy wasted through muscle use. The minimum work required is then the difference in gravitational potential energy the hiker gains. This ensures that, in an ideal world without inefficiencies, all your effort purely converts to the potential energy increase.
In real-world applications, the work-energy principle is a vital tool for predicting energy requirements for physical activities and assessing the basic relationship between work and energy under conservative forces like gravity.

Physics Problems

Physics problems, like the one involving the hiker, often combine various principles and require critical thinking to solve. For problems involving gravitational potential energy and work, clarity in understanding the system is crucial.
Consider these helpful tips:

• Read Carefully: Always start by carefully analyzing the problem statement to determine all given data and what needs to be calculated.
• Identify Key Variables: Note down variables like mass, height, and gravitational acceleration, as they are crucial for potential energy calculations.
• Apply Appropriate Formulas: Use the correct formulas, ensuring you understand their derivation and applicability.
• Consider All Forces: Think about additional forces or factors that affect work done, like friction or air resistance.

These approaches help break down future complicated-looking physics problems into simpler, manageable tasks, guided by fundamental concepts like potential energy and work.