Problem 21 - Applied Physics

Question

A painter weighing \(630 \mathrm{~N}\) climbs to a height of \(5.00 \mathrm{~m}\) on a ladder. (a) How much work does she do in climbing the ladder? (b) What is the increase in gravitational potential energy of the painter? (c) Where does the energy come from to cause this increase in potential energy?

Step-by-Step Solution

Answer

(a) 3150 J, (b) 3150 J, (c) The energy comes from the painter's food-based chemical energy.

Step 1: Understand the Work Concept

Work done is defined as the product of the force applied and the distance over which the force is applied in the direction of the force. The formula is given by \( W = F \cdot d \cdot \cos{\theta} \), where \( F \) is force, \( d \) is the distance, and \( \theta \) is the angle between the force and the direction of movement.

Step 2: Calculate Work Done in Climbing the Ladder

In this scenario, the force applied is equal to the weight of the painter, \( 630 \mathrm{~N} \), and the height (distance) \( d \) climbed is \( 5.00 \mathrm{~m} \). The force and movement are in the same direction (vertically upwards), so \( \theta = 0 \) and \( \cos{0} = 1 \). The work done is \( W = 630 \mathrm{~N} \times 5.00 \mathrm{~m} \times 1 = 3150 \mathrm{~J} \).

Step 3: Understanding Gravitational Potential Energy

Gravitational potential energy \( U \) is given by \( U = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is height. However, using weight \( mg \), the increase in gravitational potential energy \( \Delta U \) is exactly equivalent to the work done: \( \Delta U = 630 \mathrm{~N} \times 5.00 \mathrm{~m} = 3150 \mathrm{~J} \).

Step 4: Identify the Source of Energy for Climbing

The energy required to increase the gravitational potential energy comes from the chemical energy stored in the painter's body, specifically from the energy stored in her muscles gained from food.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is an important concept in physics. It describes the energy an object possesses due to its position in a gravitational field. The higher an object is above the ground, the more gravitational potential energy it has. This energy is calculated using the formula: \[ U = mgh \] where:

• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \) on Earth),
• \( h \) is the height of the object above the reference point.

In our example, the painter increases her gravitational potential energy by climbing to a height of 5 meters. This increase is exactly the work done against gravity to lift her body up the ladder.

Work Done by Forces

Work is a measure of energy transfer when a force is applied to an object, causing it to move. The basic formula for calculating work is:\[ W = F \cdot d \cdot \cos{\theta} \]Here, \( F \) is the force applied, \( d \) is the displacement of the object in the direction of the force, and \( \theta \) is the angle between the force and the direction of displacement. In the case of our painter:

• The force is her weight, which is \( 630 \, \text{N} \).
• She climbs a vertical height (displacement) of \( 5 \, \text{m} \).
• Since the force and the displacement are in the same direction (vertically upwards), \( \theta = 0 \), making \( \cos{0} = 1 \).

By plugging these values into the formula, the work done by the painter is calculated as \( 3150 \, \text{J} \). This work is directly related to the increase in her gravitational potential energy.

Conservation of Energy

The law of conservation of energy is a fundamental principle in physics. It states that energy cannot be created or destroyed; it can only be transformed from one form to another. In our scenario with the painter, the chemical energy stored in her body is transformed into mechanical energy as she climbs the ladder. This mechanical energy does work against gravity, increasing her gravitational potential energy. This concept is crucial because it helps us understand that the energy spent by the painter is not lost but merely changes form, emphasizing the balance of energy in closed systems. The energy she uses for climbing comes from the food she consumes which is stored as chemical energy in her muscles. Thus, it affirms the conservation of energy law as energy shifts from chemical to potential form.

Physics Problem Solving

Solving physics problems often requires a systematic approach. In addressing our example with the painter, we can apply a series of logical steps:

• **Identify the known variables**: Here, it's the weight of the painter and the height she climbs.
• **Select the right formulas**: Use formulas related to work (\( W = F \cdot d \cdot \cos{\theta} \)) and gravitational potential energy (\( U = mgh \)).
• **Substitute these values into the equations**: Plug in the known values to find the solution.
• **Verify units and check calculations**: An important step to ensure the calculated figures make sense.

By following these steps, students can approach and solve physics problems more confidently. Practicing this structured approach simplifies complex scenarios, making the process of learning physics both analytical and rewarding.