Problem 55 - Master Resource Book in JEE Main Physics

Question

An engine pumps water through a hose pipe. Water passes through the pipe and leaves to with a velocity of \(2 \mathrm{~m} / \mathrm{s}\). The mass per unit length of water in the pipe is \(100 \mathrm{~kg} / \mathrm{m}\). What is the power of the engine? [CBSE PMT 2010] (a) \(800 \mathrm{~W}\) (b) \(400 \mathrm{~W}\) (c) \(200 \mathrm{~W}\) (d) \(100 \mathrm{~W}\)

Step-by-Step Solution

Answer

The power of the engine is \(400 \, \text{W}\).

Step 1: Identify Given Variables

We have the velocity of water, \(v = 2 \, \text{m/s}\), and the mass per unit length of water in the pipe, \(\mu = 100 \, \text{kg/m}\). These will be used to find the power of the engine.

Step 2: Calculate the Mass Flow Rate

The mass flow rate (\( \dot{m} \)) is given by the product of the mass per unit length (\( \mu \)) and the velocity (\( v \)). \[ \dot{m} = \mu \times v = 100 \, \text{kg/m} \times 2 \, \text{m/s} = 200 \, \text{kg/s} \]

Step 3: Calculate the Kinetic Energy of Water Per Second

The kinetic energy per unit time, or power (\( P \)), is given by \( P = \frac{1}{2} \dot{m} v^2 \). Substitute the values we have: \[ P = \frac{1}{2} \times 200 \, \text{kg/s} \times (2 \, \text{m/s})^2 = \frac{1}{2} \times 200 \times 4 = 400 \, \text{W} \]

Key Concepts

Kinetic Energy

Kinetic energy (KE) is the energy that an object possesses due to its motion. It plays a crucial role in various physics calculations, particularly when dealing with moving fluids like water in a pipe. To understand kinetic energy, remember that it depends on two main factors:

• Mass of the object or substance.
• Velocity at which it moves.

The expression for kinetic energy is given by the formula:\[ KE = \frac{1}{2} mv^2 \]where \( m \)is the mass of the water and \( v \)is its velocity. When water moves through a pipe at a certain speed, its kinetic energy can be converted into work, such as turning a turbine or driving a piston.
In our exercise, the power calculation involves finding out how much kinetic energy is imparted to the flowing water every second. This helps explain why the mass flow rate and the velocity are crucial for determining the engine's power. Thus, the power output can be equated to the rate at which the water's kinetic energy changes.

Mass Flow Rate

Mass flow rate (\dot{m}) is a measure of the mass of a substance passing through a given surface per unit of time. In the context of fluid dynamics, it allows us to understand how much fluid is moving through a pipe at any given time, which is essential for calculating the power needed by an engine to maintain the flow.
The mass flow rate is calculated using the formula:\[ \dot{m} = \mu \times v \]where \( \mu \)is the mass per unit length of the water in the pipe, and \( v \)is the velocity of the water. In our exercise, the mass flow rate amounts to 200 kg/s, indicating that each second, 200 kg of water flows through the pipe. Understanding this concept is vital because it indicates the workload the engine must handle, influencing how much power is needed to maintain the fluid's velocity within the system.

Velocity of Fluid

Velocity is a vector quantity that describes the speed of a fluid in a given direction. The velocity of the fluid is crucial for energy and power calculations because it significantly impacts the energy conversion efficiency and the rate at which energy is transferred.
When we talk about the velocity of fluid in pipes, we're examining how fast the fluid particles move, which helps us determine kinetic energy and mass flow rates.
In the given exercise, the water exits the pipe with a velocity of 2 m/s. This velocity helps determine the kinetic energy of the water, which corresponds to the power output as described by the engine. Higher fluid velocity typically means increased kinetic energy, translating into a greater requirement for power generation by the system. Recognizing how velocity affects fluid dynamics problems aids in deepening the understanding of complex energy systems.