We said in Section 9.1 that the potential energy of water at the top of a dam or waterfall is converted into heat when the water dashes against rocks at the bottom. The potential energy of the water at the top is equal to EP = mgh, where m is the mass of the water, g is the acceleration of the falling water due to gravity 1g = 9.81 m>s22, and h is the height of the water. Assuming that all the energy is converted to heat, calculate the temperature rise of the water in degrees Celsius after falling over California's Yosemite Falls, a distance of 739 m. The specific heat of water is 4.18 J/(g·K).
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First, calculate the potential energy of the water using the formula EP = mgh. Here, m is the mass of the water, g is the acceleration due to gravity (9.81 m/s^2), and h is the height of the waterfall (739 m).
Next, convert the potential energy into heat energy. Since the problem assumes that all the potential energy is converted into heat, the heat energy is equal to the potential energy.
Then, use the formula for heat transfer q = mcΔT to calculate the temperature change ΔT. Here, q is the heat energy, m is the mass of the water, c is the specific heat of water (4.18 J/(g·K)), and ΔT is the temperature change.
Rearrange the formula to solve for ΔT: ΔT = q / (mc). Substitute the values for q, m, and c into the formula.
Finally, note that the temperature change will be in Kelvin. To convert it to degrees Celsius, remember that a change of 1 K is equivalent to a change of 1°C. Therefore, the temperature change in degrees Celsius is the same as in Kelvin.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Potential Energy
Potential energy is the energy stored in an object due to its position in a gravitational field. In the context of water at the top of a dam, it is calculated using the formula EP = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. This energy is converted into kinetic energy as the water falls, and ultimately into heat energy upon impact.
Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). For water, this value is 4.18 J/(g·K), indicating that it takes 4.18 joules of energy to raise the temperature of one gram of water by one degree. This concept is crucial for calculating the temperature change of water after it has converted potential energy into heat.
The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the potential energy of the falling water is transformed into thermal energy (heat) upon impact. Understanding this principle allows us to calculate the resulting temperature increase of the water after it falls, as all potential energy is assumed to convert into heat energy.
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