Problem 32 - University Physics with Modern Physics

Question

While painting the top of an antenna 225 \(\mathrm{m}\) in height, a worker accidentally lets a \(1.00-\mathrm{L}\) wattle 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?

Step-by-Step Solution

Answer

The water's temperature increases by approximately 0.527°C.

Step 1: Understand the Problem

The problem involves a change in mechanical energy, specifically gravitational potential energy, as the water bottle falls from 225 meters to the ground. This change in mechanical energy is converted into heat, which then increases the temperature of the water in the bottle.

Step 2: Calculate the Change in Mechanical Energy

Mechanical energy change is equal to the change in gravitational potential energy, which is calculated using the formula: \[ \Delta E = mgh \] where \( m \) is the mass of the water, \( g \) is the acceleration due to gravity (\(9.81\, \text{m/s}^2\)), and \( h \) is the height (225 m). Since the density of water is \(1\, \text{kg/L}\), the mass \( m \) of 1 L of water is 1 kg. Thus, \[ \Delta E = 1 \times 9.81 \times 225 = 2207.25 \text{ J} \].

Step 3: Relate to Temperature Change

The heat energy transferred to the water \( Q \) is equal to the change in mechanical energy calculated. The increase in temperature \( \Delta T \) can be found using the specific heat capacity formula: \[ Q = mc\Delta T \] where \( c \) is the specific heat capacity of water \(4.186 \text{ J/g°C}\), and \( m \) is 1000 g (since 1 kg = 1000 g). Solving for \( \Delta T \):\[ \Delta T = \frac{Q}{mc} = \frac{2207.25}{1000 \times 4.186} = 0.527 \text{°C} \].

Key Concepts

Gravitational Potential Energy

Gravitational Potential Energy (GPE) is a form of energy an object possesses because of its position in a gravitational field. Essentially, it's the energy stored in an object as it is positioned above a reference level, such as the ground. The formula to calculate the gravitational potential energy is \[ GPE = mgh \]. Here, \( m \) represents mass, \( g \) is the acceleration due to gravity (approximately \(9.81\, \text{m/s}^2\) on Earth), and \( h \) stands for the height above the reference level. The GPE highlights two essential components:

• **Mass of the Object**: A heavier object contains more potential gravitational energy when at the same height as a lighter one.
• **Height of the Object**: The higher the position of the object above the ground, the more gravitational potential energy it possesses.

When a falling object reaches the ground, its gravitational potential energy is converted largely to other forms of energy, such as heat. This transformation of energy is seen in the exercise, where the energy change contributes to an increase in the water bottle's temperature.

Temperature Change

Temperature change in an object happens when energy is absorbed or released, influencing the object's thermal state. The rise or fall in temperature depends largely on the heat energy transferred, the object's mass, and its specific heat capacity. In the context of the exercise, the temperature change occurs as the gravitational potential energy is converted into heat when the water bottle hits the ground. The amount of temperature shift \( \Delta T \) is determined using the formula:\[ \Delta T = \frac{Q}{mc} \]Here, \( Q \) is the heat energy, \( m \) is the mass, and \( c \) symbolizes the specific heat capacity. A clear understanding of this equation helps in discerning how external factors, like energy transfer, impact temperature variations. In many scenarios:

• **Higher the energy transfer**, the more significant the temperature change will be.
• **Mass of the object**: More massive objects experience less temperature change for the same amount of energy transferred.

This principle is crucial in understanding the energy transformations in both everyday occurrences and more complex scientific processes.

Specific Heat Capacity

Specific heat capacity is a property that describes the amount of heat required to change a substance's temperature by a given amount, typically 1°C. It's a pivotal factor in determining how substances react to thermal energy. In the equation \( Q = mc\Delta T \), \( c \) stands for the specific heat capacity, which plays a significant role in calculating temperature change. The specific heat capacity of water is notably high at \(4.186 \text{ J/g°C}\), meaning it takes more energy to alter water's temperature compared to many other substances. Key insights about specific heat capacity include:

• **Substances with higher specific heat** efficiently resist temperature changes when energy is added or removed.
• **Water's high specific heat capacity** gives it excellent thermal stability, making it essential in regulating climates and supporting life.

Understanding specific heat helps in numerous practical scenarios, like cooking, where knowing how a material will respond to heating can guide temperature control for optimal results.