Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 17: Temperature and Heat

Problem 29

Chapter 17, Problem 29

While painting the top of an antenna 225 m in height, a worker accidentally lets a 1.00-L water bottle fall from his lunchbox. The bottle lands in some bushes at ground level and does not break. If a quantity of heat equal to the magnitude of the change in mechanical energy of the water goes into the water, what is its increase in temperature?

Verified step by step guidance

Identify the initial and final states of the water bottle. Initially, the bottle is at a height of 225 m, and finally, it is at ground level. The change in mechanical energy is due to the change in gravitational potential energy.

Calculate the change in gravitational potential energy (ΔPE) using the formula: ΔPE = m * g * h, where m is the mass of the water, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height (225 m).

Since the volume of the water is 1.00 L, convert this volume to mass. Assuming the density of water is 1 kg/L, the mass m is 1 kg.

The problem states that the change in mechanical energy is converted into heat energy, which increases the temperature of the water. Use the formula for heat transfer: Q = m * c * ΔT, where Q is the heat energy (equal to the change in potential energy), c is the specific heat capacity of water (approximately 4.18 J/g°C), and ΔT is the change in temperature.

Rearrange the heat transfer formula to solve for the change in temperature: ΔT = Q / (m * c). Substitute the values for Q (from step 2), m (1 kg), and c (4.18 J/g°C) to find the increase in temperature of the water.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mechanical Energy

Mechanical energy is the sum of potential energy and kinetic energy in a system. In this scenario, the potential energy of the water bottle at the top of the antenna is converted into kinetic energy as it falls. The change in mechanical energy is the difference between the initial potential energy and the final kinetic energy when the bottle reaches the ground.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. As the water bottle falls, its potential energy is converted into kinetic energy, and upon impact, this energy is transformed into heat, increasing the temperature of the water. Understanding this transformation is crucial for calculating the temperature change.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It is essential for determining how much the temperature of the water increases when the mechanical energy is converted into heat. The specific heat capacity of water is typically 4.18 J/g°C, which helps in calculating the temperature rise from the energy absorbed.

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