Problem 20 - Applied Physics

Question

A window washer with mass \(90.0 \mathrm{~kg}\) first climbs \(45.0 \mathrm{~m}\) upward to the top of a building, then from the top goes down \(85.0 \mathrm{~m}\) to the ground. (a) What is the potential energy of the window washer at the top of the building, using his initial position as the reference level? (b) Find the potential energy of the window washer at ground level with respect to his initial position.

Step-by-Step Solution

Answer

(a) 39690 J; (b) -35244 J

Step 1: Understand the Concept of Gravitational Potential Energy

Gravitational potential energy (PE) is given by the formula: \( PE = mgh \), where \( m \) is the mass in kilograms, \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \) on Earth), and \( h \) is the height in meters above the reference level. The reference level is where we consider the potential energy to be zero.

Step 2: Calculate Potential Energy at the Top of the Building

First, identify the mass of the window washer as \( 90.0 \text{ kg} \) and the height climbed as \( 45.0 \text{ m} \). Using the formula \( PE = mgh \), we calculate:\[ PE = 90.0 \times 9.81 \times 45.0 \]Calculate the product to find the potential energy at the top.

Step 3: Perform the Calculation

Substitute the values into the potential energy formula:\[ PE = 90.0 \times 9.81 \times 45.0 = 39690 \text{ J} \]Therefore, the potential energy at the top of the building is \( 39690 \text{ J} \).

Step 4: Calculate Potential Energy at Ground Level

The window washer descends \( 85.0 \text{ m} \) from the top, reaching \( 45.0 \text{ m} - 85.0 \text{ m} = -40.0 \text{ m} \) relative to the initial position, which means he is 40.0 m below the initial position. Use the potential energy formula again:\[ PE = 90.0 \times 9.81 \times (-40.0) \]

Step 5: Perform the Calculation for Ground Level

Substitute the values into the formula for the potential energy at the ground level:\[ PE = 90.0 \times 9.81 \times (-40.0) = -35244 \text{ J} \]Thus, the potential energy at the ground level is \( -35244 \text{ J} \).

Key Concepts

Mass

Mass is a fundamental property of matter that represents the amount of matter in an object. It is measured in kilograms (kg) and remains constant regardless of its location. In our scenario with the window washer, the mass is given as 90.0 kg. This mass is used to calculate gravitational potential energy.
The mass of an object affects how much gravitational pull it experiences.

• A larger mass means more gravitational force is exerted on the object.
• This also implies a higher potential energy when elevated, given everything else is constant.

This consistency in mass, regardless of position, makes it a reliable factor in potential energy calculations.

Acceleration due to Gravity

The acceleration due to gravity is a key factor in gravitational potential energy calculations. It is approximately 9.81 meters per second squared (m/s²) on Earth. This constant represents the rate at which objects accelerate towards the Earth's center when dropped.
It is crucial to understand this principle when calculating potential energy as it directly impacts the amount of energy an object has when raised to a height.

• Every kilogram of mass experiences a force due to gravity.
• This force is directly proportional to the value of 9.81 m/s² in typical physics problems.

Since this is a constant value, it simplifies calculations, allowing us to focus on variables like height and mass in potential energy equations.

Height

Height is a vital parameter when calculating gravitational potential energy, denoted by the variable \( h \). It is the vertical distance from the reference level, which affects how much potential energy an object possesses.
In this exercise, the window washer climbs 45.0 meters initially and then descends further. The height determines the change in potential energy.

• The higher the object, the more potential energy it has.
• This is because gravitational energy increases as you move against the force of gravity.

Thus, understanding and calculating heights accurately affect the result in forming the gravitational potential energy calculations.

Reference Level

The reference level is an arbitrary point where potential energy is considered zero. It serves as the baseline to measure an object's height from. In gravitational potential energy problems, selecting the reference level is crucial.
In the window washer's scenario, his initial position is used as the reference level for measuring height.

• The starting point or reference level can impact the sign of the potential energy.
• Above the reference level results in positive potential energy, whereas below it results in negative.

This reference helps define how height is measured and influences the resulting potential energy calculations. It emphasizes that potential energy can depend on relative position as much as the physical properties of the object.