Problem 58 - Master Resource Book in JEE Main Physics

Question

Water falls from a height of \(60 \mathrm{~m}\) at the rate of \(15 \mathrm{~kg} / \mathrm{s}\) to operate a turbine. The losses due to frictional force are \(10 \%\) of energy. How much power is generated by the turbine \(\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) ? [CBSE PMT 2008] (a) \(12.3 \mathrm{~kW}\) (b) \(7.0 \mathrm{~kW}\) (c) \(8.1 \mathrm{~kW}\) (d) \(10.2 \mathrm{~kW}\)

Step-by-Step Solution

Answer

The power generated is 8.1 kW (option c).

Step 1: Understand the Problem

We need to calculate the power generated by the turbine using the data provided. The energy loss due to frictional forces is a notable factor to consider. We must apply the physics concepts of gravitational potential energy and power, considering energy losses.

Step 2: Calculate the Potential Energy of Water

The potential energy (PE) of water falling from a height is calculated using the formula: \[PE = m \cdot g \cdot h\]where:- \(m = 15\, \text{kg/s}\) is the mass flow rate,- \(g = 10\, \text{m/s}^2\) is the gravitational acceleration,- \(h = 60\, \text{m}\) is the height.Plug in the values:\[PE = 15 \times 10 \times 60 = 9000 \, \text{J/s}\]This is the energy rate or power without considering losses.

Step 3: Account for Energy Losses

The energy losses due to friction are \(10\%\) of the total potential energy. To find the energy that effectively operates the turbine, subtract this loss:\[\text{Energy loss} = 0.10 \times 9000 = 900\, \text{J/s}\]Thus, the useful power is:\[\text{Useful power} = 9000 - 900 = 8100 \, \text{J/s}\]Note that 1 J/s is equivalent to 1 watt, so this is 8100 watts or 8.1 kW.

Step 4: Choose the Correct Answer Option

The calculated power generated by the turbine (accounting for energy losses) is \(8.1\, \text{kW}\). Comparing with the options given, the correct answer is (c) \(8.1\, \text{kW}\).

Key Concepts

Potential Energy

Potential energy is a form of energy that an object possesses due to its position or configuration. For water falling from a height to operate a turbine, the potential energy is determined by its elevation and mass. This type of energy is called gravitational potential energy because it depends on the gravitational force acting on the mass.

The potential energy of an object can be calculated using the formula:

• \( PE = m \cdot g \cdot h \)

where:

• \(m\) is the mass in kilograms,
• \(g\) is the acceleration due to gravity, typically \(10 \, \text{m/s}^2\),
• \(h\) is the height in meters.

By understanding and applying this formula, we can easily calculate the potential energy of water, which is crucial for determining how much power a turbine can generate.

Power Generation

Power generation is the process of converting various forms of energy into electric power. In this exercise, the primary energy form is potential energy from falling water, which the turbine converts into electrical energy. To find power generation, we use the rate of energy conversion or energy used per unit of time.

Power can be calculated as:

• \( P = \frac{E}{t} \)

where:

• \(E\) is energy in joules,
• \(t\) is time in seconds.

In the exercise, power is determined directly from potential energy per second (also known as joules per second), which outputs in watts (1 J/s = 1 W). With water flow providing potential energy, power generation is effectively the conversion of gravitational potential energy into electric energy by the turbine.

Frictional Force Losses

Frictional force losses refer to the energy lost due to frictional forces that oppose motion. These losses are inevitable in mechanical systems, like the turbine in this exercise.

In the context of the power generated by these systems:

• Frictional losses deplete a portion of the otherwise available energy.
• This energy is often lost as heat.

For the exercise, frictional losses amount to 10% of the total potential energy. Hence, only 90% of the initial energy contributes to actual power generation. Calculating effective usable energy involves these steps:

• Find the loss: \(0.10 \times 9000 = 900 \, \text{J/s}\).
• Subtract this from total potential energy: \(9000 - 900 = 8100 \, \text{J/s}\).

Understanding and addressing these losses is crucial in predicting how efficiently a power system like a turbine operates.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object as a result of its vertical position or height. The formula for gravitational potential energy is given by:

• \( PE = m \cdot g \cdot h \)

This represents how much work gravitational force can perform due to the object's height. In power systems, like a water-falling turbine, this energy becomes crucial because it is harnessed directly for generating electricity.

Working through this exercise:

• The energy is calculated based on the water's flow rate and the height it falls from.
• Gravitational acceleration is needed to determine how much force is acting on the water mass.

The usefulness of gravitational potential energy lies in its ability to be transformed into kinetic energy, then mechanical energy, and eventually into electrical energy through turbines, providing eco-friendly power solutions.