Problem 10 - Master Resource Book in JEE Main Physics

Question

Water falls from a height of \(500 \mathrm{~m}\). What is the rise in temperature of water at the bottom if whole energy is used up in heating water ? (a) \(0.96^{\circ} \mathrm{C}\) (b) \(1.02^{\circ} \mathrm{C}\) (c) \(1.16^{\circ} \mathrm{C}\) (d) \(0.23^{*} \mathrm{C}\)

Step-by-Step Solution

Answer

The rise in temperature is \(1.16^{\circ}\text{C}\) (option c).

Step 1: Understand the Concepts Involved

Water falling from a height has potential energy which is converted into heat energy when it hits the ground. The question asks us to find out how much the temperature of the water increases as a result of this energy conversion.

Step 2: Identify Given Values and Constants

The height from which water falls is given as \(500 \mathrm{~m}\). The acceleration due to gravity \(g\) is \(9.8 \mathrm{~m/s^2}\). The specific heat capacity of water \(c\) is \(4.186 \mathrm{~J/g^ \circ C}\). Assume 1 gram of water for simplicity.

Step 3: Calculate Potential Energy

Potential energy (PE) of the water is calculated using the formula:\[ \text{PE} = mgh \]where \(m\) is the mass of water (assumed as \(1\mathrm{~g}\)), \(g\) is the acceleration due to gravity, and \(h\) is the height. As \(m\) is \(1 \mathrm{~g}\), the potential energy is:\[ \text{PE} = 1 \times 9.8 \times 500 = 4900 \mathrm{~J} \]

Step 4: Use Energy Conversion to Find Temperature Rise

The potential energy of \(4900 \mathrm{~J}\) gets converted into heat energy \(Q\) which increases the water's temperature. Use the specific heat capacity formula:\[ Q = mc\Delta T \]where \(\Delta T\) is the change in temperature. Rearrange to find \(\Delta T\):\[ \Delta T = \frac{Q}{mc} = \frac{4900 \mathrm{~J}}{1 \mathrm{~g} \times 4.186 \mathrm{~J/g^\circ C}} \approx 1.17^\circ \mathrm{C} \]This approximates to \(1.16^\circ \mathrm{C}\).

Step 5: Select the Correct Option

Match the calculated temperature change with the given options. The closest match:Option (c) \(1.16^{\circ} \mathrm{C}\).

Key Concepts

Potential Energy

When water is at a height, it possesses potential energy due to its position. This energy results from the gravitational pull of the Earth. The higher the object, the more potential energy it has.
For water falling from a height of 500 meters, the potential energy is given by the formula:

• \[ \text{PE} = mgh \]
• Where \( m \) is the mass, \( g \) is the acceleration due to gravity (9.8 \( \mathrm{m/s^2} \)), and \( h \) is the height.

When the water falls and hits the ground, this stored energy is released.This concept of potential energy helps us understand how energy is converted from one form to another, a crucial principle in physics.
In this specific scenario, the potential energy of the water is converted into heat.

Specific Heat Capacity

The specific heat capacity is a measure of how much heat energy a substance can store per unit mass before its temperature changes.
For water, this value is \( 4.186 \mathrm{~J/g^\circ C} \). This means it takes 4.186 joules of energy to raise the temperature of 1 gram of water by 1 degree Celsius.Understanding specific heat capacity is crucial for solving problems related to temperature changes, as it tells us how efficiently a substance can store heat energy.

• A higher specific heat capacity means the substance can store more energy before its temperature rises.
• For water, this makes it an excellent medium for storing and absorbing heat.

When energy, such as the potential energy from falling water, is converted into heat, this constant helps us determine how much the temperature of the water will increase.

Temperature Rise

The change in temperature of a substance occurs when it absorbs or releases heat energy.
In the case of falling water, the potential energy is converted entirely into heat energy upon impact with the ground, leading to a temperature rise.We can calculate this rise using the specific heat capacity formula:

• \[ Q = mc\Delta T \]

Here:

• \( Q \) is the heat energy (4900 J in this case),
• \( m \) is the mass of the water (1 gram),
• \( c \) is the specific heat capacity,

By rearranging the formula, \( \Delta T = \frac{Q}{mc} \), we find the increase in water temperature to be approximately 1.17°C.
Thus, through energy conversion, the potential energy causes the water's temperature to rise, showing the direct relationship between energy and temperature change.