Problem 30 - College Physics

Question

\(\bullet\) Much of the energy of falling water in a waterfall is converted into heat. If all the mechanical energy is converted into heat that stays in the water, how much of a rise in temperature occurs in a 100 m waterfall?

Step-by-Step Solution

Answer

The temperature of the water rises by approximately 2.34°C.

Step 1: Understanding the Problem

We need to find the rise in temperature of water falling from a height of 100 m, assuming all mechanical energy is converted into heat. We can use the principle of conservation of energy, where the potential energy lost is equal to the heat energy gained.

Step 2: Calculate Potential Energy Loss

The potential energy (PE) lost by the water can be calculated using the formula: \(PE = mgh\), where \(m\) is the mass of water, \(g = 9.8\, \text{m/s}^2\) is the acceleration due to gravity, and \(h = 100\, \text{m}\) is the height of the fall.

Step 3: Calculate Heat Energy Gained

The heat energy (Q) gained by the water can be calculated using the formula: \(Q = mc\Delta T\), where \(m\) is the mass of water, \(c = 4.18\, \text{J/g}^\circ\text{C}\) is the specific heat capacity of water, and \(\Delta T\) is the change in temperature we wish to find.

Step 4: Equate and Solve for Temperature Change

Since all potential energy is converted into heat, set the potential energy equal to the heat energy: \(mgh = mc\Delta T\). The masses cancel out, simplifying to \(gh = c\Delta T\). Rearranging gives \(\Delta T = \frac{gh}{c}\).

Step 5: Calculate Temperature Change

Substitute the known values into the equation: \(\Delta T = \frac{9.8 \times 100}{4.18}\). Calculate \(\Delta T\) to find the rise in temperature.

Step 6: Result: Temperature Rise

After performing the calculation, \(\Delta T = \frac{9.8 \times 100}{4.18} \approx 2.34^\circ\text{C}\). Thus, the temperature rise of the water is approximately 2.34°C.

Key Concepts

Potential Energy

Potential energy is the stored energy that an object has due to its position or height relative to a reference point. In the case of a waterfall, the water at the top has potential energy due to its height above the ground.

This energy is calculated using the formula:
\[ PE = mgh \]
where:

• \( m \) is the mass of the water,
• \( g \) is the acceleration due to gravity \((9.8 \, \text{m/s}^2)\),
• \( h \) is the height from which the water falls (100 m in this exercise).

During the descent, this potential energy is converted into other forms of energy, such as kinetic energy and subsequently heat energy when the water reaches the bottom. Understanding potential energy helps us appreciate the initial amount of energy available for conversion.

Specific Heat Capacity

Specific heat capacity is a property of a material that tells us how much heat is needed to raise the temperature of 1 gram of the material by 1 degree Celsius. For water, we often use specific heat capacity in the form of:

\[ c = 4.18 \text{ J/g}^\circ\text{C} \]
This means it takes 4.18 Joules of energy to increase the temperature of 1 gram of water by 1°C. Knowing the specific heat capacity is crucial when calculating the temperature change in substances due to energy conversion.

In the problem with the waterfall, the specific heat capacity allows us to relate the heat energy gained by water to its temperature change. This property is pivotal in determining how different substances respond to heating.

Temperature Change

When energy is converted into heat, it can cause a change in temperature of a substance. In this exercise, the waterfall's potential energy is converted into heat energy, which increases the temperature of the water. The change in temperature \( \Delta T \) is given by:

\[ Q = mc\Delta T \]
Rearranging this formula gives us:
\[ \Delta T = \frac{Q}{mc} = \frac{gh}{c} \]
where the mass \( m \) cancels out because it appears on both sides of the equation.

After calculating, we find \( \Delta T \approx 2.34^\circ\text{C} \). This means the energy conversion results in the water’s temperature rising by about 2.34 degrees Celsius. This demonstrates the link between energy conversion and observable temperature changes.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In the waterfall scenario, all of the energy initially present as potential energy is conserved and transformed into heat energy.

The transformation sequence might look like this:

• Water starts at the top with a certain amount of potential energy.
• As it falls, this energy converts into kinetic energy (energy of movement).
• Upon reaching the bottom, the kinetic energy transforms into heat, causing an increase in water temperature.

Thus, the energy equation \( mgh = mc\Delta T \) illustrates conservation, as it relates the initial potential energy to the resultant heat energy that causes a temperature rise. Understanding this concept helps us apply the conservation principle to various physical scenarios seamlessly.